Mathematics Form 1-4 : CHAPTER FIFTY FOUR – PROBABILITY

Specific Objectives

By the end of the topic, the learner should be able to:

• Define probability;
• Determine probability from experiments and real-life situations;
• Construct a probability space;
• Determine theoretical probability;
• Differentiate between discrete and continuous probability;
• Differentiate mutually exclusive and independent events;
• State and apply laws of probability;
• Use a tree diagram to determine probabilities.

Content

• Probability
• Experimental probability
• Range of probability measure 0 ≤ P(x) ≤ 1
• Probability space
• Theoretical probability
• Discrete and continuous probability (simple cases only)
• Combined events (mutually exclusive and independent events)
• Laws of probability
• Tree diagrams

Introduction

Probability is the likelihood of the occurrence of an event or the numerical measure of chance. It quantifies how likely an event is to happen.

Experimental Probability

Experimental probability is determined by experience or experiment. The process or observation is called the experiment. Each attempt is called a trial, and the result of a trial is the outcome. The experimental probability of a result is given by the ratio of the number of favorable outcomes to the total number of trials.

Example

A boy had a fair die with faces marked 1 to 6. He threw this die 50 times and recorded the number on the top face each time. The results of his experiment are shown below.

What is the experimental probability of getting:

a) 1 b) 6

Solution

a) P(Event) = P(1) = 11/50

b) P(6) = 8/50

Example

From past records, out of ten matches a school football team played, it won seven. How many possible games might the school win in thirty matches?

Solution

P(winning in one match) = 7/10.

Therefore, the number of possible wins in thirty matches = 7/10 × 30 = 21 matches.

Range of Probability Measure

If P(A) is the probability of an event A happening and P(A’) is the probability of event A not happening, then P(A’) = 1 – P(A) and P(A’) + P(A) = 1.

Probabilities are expressed as fractions, decimals, or percentages.

Probability Space

A list of all possible outcomes is called the probability space or sample space. For example, when tossing a coin, the outcomes head or tail have equal chances of occurring. Such events are said to be equally likely or equiprobable.

Theoretical Probability

Theoretical probability can be calculated without using past experience or experiments. It is given by the ratio of the number of favorable outcomes to the total number of outcomes.

Example

A basket contains 5 red balls, 4 green balls, and 3 blue balls. If a ball is picked at random from the basket, find:

a) The probability of picking a blue ball

b) The probability of not picking a red ball

Solution

a) Total number of balls is 12.

The number of blue balls is 3.

Therefore, P(blue ball) = 3/12.

b) The number of balls which are not red is 7.

Therefore, P(not a red ball) = 7/12.

Example

A bag contains 6 black balls and some brown ones. If a ball is picked at random, the probability that it is black is 0.25. Find the number of brown balls.

Solution

Let the total number of balls be x.

The probability of picking a black ball is 6/x.

Therefore, 6/x = 0.25.

x = 24.

The total number of balls is 24.

The number of brown balls is 24 – 6 = 18.

Note: When all possible outcomes are countable, they are said to be discrete.

Types of Probability

Combined Events

These are probabilities of two or more events occurring together.

Mutually Exclusive Events

Occurrence of one event excludes the occurrence of the other. If A and B are two mutually exclusive events, then P(A or B) = P(A) + P(B). For example, when a coin is tossed, the result will either be a head or a tail.

Example

If a coin is tossed:

P(head) + P(tail) = 1

Note: If [OR] is used, then we add probabilities.

Independent Events

Two events A and B are independent if the occurrence of A does not influence the occurrence of B and vice versa. If A and B are independent events, the probability of both occurring together is the product of their individual probabilities:

P(A and B) = P(A) × P(B)

Note: When we use [AND], we multiply; this is the multiplication law of probability.

Example

A coin is tossed twice. What is the probability of getting a tail in both tosses?

Solution

The outcome of the 2^nd toss is independent of the outcome of the first.

Therefore,

P(T and T) = P(T) × P(T) = 1/2 × 1/2 = 1/4.

Example

A boy throws a fair coin and a regular tetrahedron with its four faces marked 1, 2, 3, and 4. Find the probability that he gets a 3 on the tetrahedron and a head on the coin.

Solution

These are independent events.

P(H) = 1/2, P(3) = 1/4.

Therefore,

P(H and 3) = P(H) × P(3) = 1/2 × 1/4 = 1/8.

Example

A bag contains 8 black balls and 5 white ones. If two balls are drawn from the bag, one at a time, find the probability of drawing a black ball and a white ball:

1. Without replacement
2. With replacement

Solution

1. There are two ways to get a black and a white ball: either drawing a white then a black, or drawing a black then a white. We find the two probabilities:

P(W followed by B) = P(W and B)

=

2. P(B followed by W) = P(B and W)

Note: The two events are mutually exclusive, so:

P(W followed by B) or (B followed by W) = P(W followed by B) + P(B followed by W)

= P(W and B) + P(B and W)

=

Since we are replacing, the number of balls remains 13.

Therefore,

P(W and B) =

P(B and W) =

Therefore,

P[(W and B) or (B and W)] = P(W and B) + P(B and W)

=

Example

Kamau, Njoroge, and Kariuki are practicing archery. The probability of Kamau hitting the target is 2/5, that of Njoroge hitting the target is 1/4, and that of Kariuki hitting the target is 3/7. Find the probability that in one attempt:

1. Only one hits the target
2. All three hit the target
3. None of them hits the target
4. Two hit the target
5. At least one hits the target

Solution

1. P(only one hits the target)

   = P(only Kamau hits and other two miss) = 2/5 × 3/5 × 4/7 = 6/35

   P(only Njoroge hits and other two miss) = 1/4 × 3/5 × 4/7 = 3/35

   P(only Kariuki hits and other two miss) = 3/7 × 3/5 × 3/4 = 27/140

   P(only one hits) = P(Kamau hits or Njoroge hits or Kariuki hits)

   = 6/35 + 3/35 + 27/140 = 9/20

2. P(all three hit) = 2/5 × 1/4 × 3/7 = 3/70

3. P(none hits) = 3/5 × 3/4 × 4/7 = 9/35

4. P(two hit the target) is the probability of:

   Kamau and Njoroge hit the target and Kariuki misses = 2/5 × 1/4 × 4/7

   Njoroge and Kariuki hit the target and Kamau misses = 1/4 × 3/7 × 3/5

   Kamau and Kariuki hit the target and Njoroge misses = 2/5 × 3/7 × 3/4

   Therefore, P(two hit target) = (2/5 × 1/4 × 4/7) + (1/4 × 3/7 × 3/5) + (2/5 × 3/7 × 3/4)

   = 8/140 + 9/140 + 18/140 = 1/4

5. P(at least one hits the target) = 1 – P(none hits the target)

   = 1 – 9/35 = 26/35

Note: P(one hits the target) is different from P(at least one hits the target).

Tree Diagram

Tree diagrams allow us to visualize all possible outcomes of an event and calculate their probabilities. Each branch in a tree diagram represents a possible outcome. A tree diagram representing a coin being tossed three times looks like this:

From the tree diagram, we see there are eight possible outcomes. To find the probability of a particular outcome, we multiply along the branches and add vertically where necessary.

The sum of the probabilities for any set of branches is always 1.

The probability of three heads is:

P(H H H) = 1/2 × 1/2 × 1/2 = 1/8

P(2 Heads and a Tail) = P(H H T) + P(H T H) + P(T H H)

= 1/2 × 1/2 × 1/2 + 1/2 × 1/2 × 1/2 + 1/2 × 1/2 × 1/2

= 1/8 + 1/8 + 1/8 = 3/8

Example

Bag A contains three red marbles and four blue marbles. Bag B contains 5 red marbles and three blue marbles. A marble is taken from each bag in turn.

1. What is the probability of getting a blue bead followed by a red?
2. What is the probability of getting a bead of each color?

Solution

1. Multiply the probabilities together:

P(blue and red) = 4/7 × 5/8 = 20/56 = 5/14

2. P(blue and red or red and blue) = P(blue and red) + P(red and blue)

   = 4/7 × 5/8 + 3/7 × 3/8 = 20/56 + 9/56 = 29/56

Example

The probability that Omweri goes to Nakuru is 1/4. If he goes to Nakuru, the probability that he will see a flamingo is 1/2. If he does not go to Nakuru, the probability that he will see a flamingo is 1/3. Find the probability that:

1. Omweri will go to Nakuru and see a flamingo.
2. Omweri will not go to Nakuru yet he will see a flamingo.
3. Omweri will see a flamingo.

Solution

Let N stand for going to Nakuru, N’ stand for not going to Nakuru, F stand for seeing a flamingo, and F’ stand for not seeing a flamingo.

1. P(He goes to Nakuru and sees a flamingo) = P(N and F) = P(N) × P(F) = 1/4 × 1/2 = 1/8

2. P(He does not go to Nakuru and yet sees a flamingo) = P(N’) × P(F) = P(N’ and F) = 3/4 × 1/3 = 1/4

3. P(He sees a flamingo) = P(N and F) or P(N’ and F) = P(N and F) + P(N’ and F) = 1/8 + 1/4 = 3/8

End of topic

Past KCSE Questions on the Topic

1. The probabilities that a husband and wife will be alive 25 years from now are 0.7 and 0.9 respectively. Find the probability that in 25 years time:
   1. Both will be alive
   2. Neither will be alive
   3. One will be alive
   4. At least one will be alive
2. A bag contains blue, green, and red pens of the same type in the ratio 8:2:5 respectively. A pen is picked at random without replacement and its colour noted.
   1. Determine the probability that the first pen picked is
      1. Blue
      2. Either green or red
   2. Using a tree diagram, determine the probability that
      1. The first two pens picked are both green
      2. Only one of the first two pens picked is red
3. A science club is made up of boys and girls. The club has 3 officials. Using a tree diagram or otherwise, find the probability that:
   1. The club officials are all boys
   2. Two of the officials are girls
4. Two baskets A and B each contain a mixture of oranges and limes, all of the same size. Basket A contains 26 oranges and 13 limes. Basket B contains 18 oranges and 15 limes. A child selected a basket at random and picked a fruit at random from it.
   1. Illustrate this information by a probabilities tree diagram
   2. Find the probability that the fruit picked was an orange.
5. In a form 1 class, there are 22 girls and boys. The probability of a girl completing the secondary education course is 3/4 whereas that of a boy is 2/3.
   1. A student is picked at random from class. Find the probability that:
      1. The student picked is a boy and will complete the course
      2. The student picked will complete the course
   2. Two students are picked at random. Find the probability that they are a boy and a girl and that both will not complete the course.
6. Three representatives are to be selected randomly from a group of 7 girls and 8 boys. Calculate the probability of selecting two girls and one boy.
7. A poultry farmer vaccinated 540 of his 720 chickens against a disease. Two months later, 5% of the vaccinated and 80% of the unvaccinated chickens contracted the disease. Calculate the probability that a chicken chosen at random contracted the disease.
8. The probability of three darts players Akinyi, Kamau, and Juma hitting the bull’s eye are 0.2, 0.3, and 1.5 respectively.
   1. Draw a probability tree diagram to show the possible outcomes
   2. Find the probability that:
      1. All hit the bull’s eye
      2. Only one of them hit the bull’s eye
      3. At most one missed the bull’s eye
9. An unbiased coin with two faces, head (H) and tail (T), is tossed three times, list all the possible outcomes. Hence determine the probability of getting:
   1. At least two heads
   2. Only one tail
10. During a certain motor rally, it is predicted that the weather will be either dry (D) or wet (W). The probability that the weather will be dry is estimated to be 7/10. The probability for a driver to complete (C) the rally during the dry weather is estimated to be 5/6. The probability for a driver to complete the rally during wet weather is estimated to be 1/10. Complete the probability tree diagram given below.

What is the probability that:

1. The driver completes the rally?
2. The weather was wet and the driver did not complete the rally?

10. There are three cars A, B, and C in a race. A is twice as likely to win as B while B is twice as likely to win as C. Find the probability that:

1. A wins the race
2. Either B or C wins the race.

11. In the year 2003, the population of a certain district was 1.8 million. Thirty per cent of the population was in the age group 15 – 40 years. In the same year, 120,000 people in the district visited the Voluntary Counseling and Testing (VCT) centre for an HIV test.

If a person was selected at random from the district in this year, find the probability that the person visited a VCT centre and was in the age group 15 – 40 years.

12. (a) Two integers x and y are selected at random from the integers 1 to 8. If the same integer may be selected twice, find the probability that:

1. |x – y| = 2
2. |x – y| is 5 or more

(iii) x > y

(b) A die is biased so that when tossed, the probability of a number r showing up is given by p(r) = Kr where K is a constant and r = 1, 2, 3, 4, 5, and 6 (the number on the faces of the die).

1. Find the value of K
2. If the die is tossed twice, calculate the probability that the total score is 11

13. Two bags A and B contain identical balls except for the colours. Bag A contains 4 red balls and 2 yellow balls. Bag B contains 2 red balls and 3 yellow balls.

1. If a ball is drawn at random from each bag, find the probability that both balls are of the same colour.
2. If two balls are drawn at random from each bag, one at a time without replacement, find the probability that:
   1. The two balls drawn from bag A or bag B are red
   2. All the four balls drawn are red

14. During inter-school competitions, football and volleyball teams from Mokagu High School took part. The probability that their football and volleyball teams would win were 3/8 and 4/7 respectively.

Find the probability that:

1. Both their football and volleyball teams win
2. At least one of their teams wins

15. A science club is made up of 5 boys and 7 girls. The club has 3 officials. Using a tree diagram or otherwise, find the probability that:

1. The club officials are all boys
2. Two of the officials are girls

16. Chicks on Onyango’s farm were noted to have either brown feathers or black tail feathers. Of those with black feathers, 2/3 were female while 2/5 of those with brown feathers were male. Otieno bought two chicks from Onyango. One had black tail feathers while the other had brown. Find the probability that Otieno’s chicks were not of the same gender.

17. Three representatives are to be selected randomly from a group of 7 girls and 8 boys. Calculate the probability of selecting two girls and one boy.

18. The probability that a man wins a game is 3/4. He plays the game until he wins. Determine the probability that he wins in the fifth round.

19. The probability that Kamau will be selected for his school’s basketball team is 1/4. If he is selected for the basketball team, then the probability that he will be selected for football is 1/3. If he is not selected for basketball, then the probability that he is selected for football is 4/5. What is the probability that Kamau is selected for at least one of the two games?

20. Two baskets A and B each contain a mixture of oranges and lemons. Basket A contains 26 oranges and 13 lemons. Basket B contains 18 oranges and 15 lemons. A child selected a basket at random and picked a fruit at random from it. Determine the probability that the fruit picked is an orange.