Lesson Worksheet: Conditional Probability Mathematics

In this worksheet, we will practice calculating conditional probability using formulas and Venn diagrams.

Q1:

Suppose 𝐴 and 𝐵 are two events. Given that 𝑃(𝐴∩𝐵)=23 and 𝑃(𝐴)=913, find 𝑃(𝐵|𝐴).

  • A913
  • B2627
  • C139
  • D23

Q2:

On a street, there are 25 houses, of which 12 have a cat and 4 have both a cat and a dog. If a house has a cat, what is the probability that there is also a dog living there? Give your answer to three decimal places.

Q3:

A bag contains 16 balls numbered from 1 to 16. To select teams for a competition, 16 students took one ball each from the bag and those whose ball had an odd number were in Team A. Calculate the probability that a student choose the number 11 given that they are in Team A.

  • A116
  • B18
  • C19
  • D111

Q4:

In the final exam, 55% of students failed chemistry, 25% failed physics and 16% failed both. What is the probability that a student passed physics given that they passed chemistry?

Q5:

Suppose that 𝐴 and 𝐵 are two events. Given that 𝑃(𝐴)=0.52 and 𝑃(𝐵|𝐴)=0.75, find 𝑃(𝐴∩𝐵).

Q6:

The probability that a student will pass an exam is 0.62. The probability that they travel abroad if they pass the exam is 0.5. What is the probability that they pass the exam and travel abroad?

Q7:

Two dice are rolled to give a pair of numbers. Given that both numbers are greater than 1, what is the probability that they are both equal to 2?

  • A225
  • B136
  • C2536
  • D125

Q8:

Suppose that 𝐴 and 𝐵 are events with probabilities 𝑃(𝐴)=0.34 and 𝑃(𝐵)=0.52. Given that 𝑃(𝐵|𝐴)=0.615, find 𝑃(𝐴∪𝐵).

Q9:

For two events 𝐴 and 𝐵, 𝑃(𝐴)=0.6, 𝑃(𝐵)=0.5, and 𝑃(𝐴∪𝐵)=0.7. Work out 𝑃(𝐴∣𝐵).

  • A15
  • B23
  • C45
  • D57
  • E67

Q10:

Suppose that 𝐴 and 𝐵 are events with probabilities 𝑃(𝐴)=0.78 and 𝑃(𝐵)=0.75. Given that 𝑃(𝐴−𝐵)=0.39, find 𝑃(𝐴∣𝐵).

Q11:

On the street, 10 houses have a cat (C), 8 houses have a dog (D), 3 houses have both, and 7 houses have neither.

Find the total number of houses on the street. Hence, find the probability that a house chosen at random has both a cat and a dog. Give your answer to three decimal places.

  • A0.167
  • B0.818
  • C0.2
  • D0.12
  • E0.136

Find the probability that a house on the street has either a cat or a dog or both. Give your answer to three decimal places.

  • A0.6
  • B0.682
  • C0.136
  • D0.318
  • E0.818

If a house on the street has a cat, find the probability that there is also a dog living there.

  • A0.136
  • B0.375
  • C0.364
  • D0.682
  • E0.3

Q12:

It has been found for two events 𝐴 and 𝐵 that 𝑃(𝐴)=0.7, 𝑃(𝐵)=0.5, and 𝑃(𝐴∪𝐵)=0.9.

Work out 𝑃(𝐴∩𝐵).

  • A920
  • B410
  • C720
  • D210
  • E310

Work out 𝑃(𝐴∣𝐵).

  • A25
  • B35
  • C37
  • D710
  • E45

Work out 𝑃(𝐵∣𝐴).

  • A35
  • B27
  • C47
  • D12
  • E37

Q13:

For two events 𝐴 and 𝐵, 𝑃(𝐴′)=0.4, 𝑃(𝐵′)=0.5, and 𝑃(𝐴′∣𝐵′)=0.2. Find 𝑃(𝐴∩𝐵).

Q14:

The probability that event 𝐴 occurs is 35. If event 𝐴 does NOT occur, then the probability of event 𝐵 occurring is 23. What is the probability that event 𝐴 does NOT occur and event 𝐵 occurs?

  • A25
  • B415
  • C35
  • D910
  • E1115

Q15:

For two independent events 𝐴 and 𝐵, where 𝑃(𝐴)=0.2 and 𝑃(𝐵)=0.3, calculate 𝑃(𝐴∣𝐵).

Q16:

Suppose 𝑃(𝐴)=25 and 𝑃(𝐵)=37. The probability that event 𝐴 occurs and event 𝐵 also occurs is 15. Calculate 𝑃(𝐴∣𝐵), and then evaluate whether events 𝐴 and 𝐵 are independent.

  • A𝑃(𝐴∣𝐵)=37;𝑃(𝐴∣𝐵)=𝑃(𝐵), so they are independent.
  • B𝑃(𝐴∣𝐵)=15;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so they are not independent.
  • C𝑃(𝐴∣𝐵)=715;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so they are not independent.
  • D𝑃(𝐴∣𝐵)=25;𝑃(𝐴∣𝐵)=𝑃(𝐴), so they are independent.
  • E𝑃(𝐴∣𝐵)=37;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so they are not independent.

Q17:

Suppose 𝑃(𝐴)=13 and 𝑃(𝐵)=16. The probability that event 𝐴 occurs and event 𝐵 also occurs is 118. Calculate 𝑃(𝐴∣𝐵), and then evaluate whether events 𝐴 and 𝐵 are independent.

  • A𝑃(𝐴∣𝐵)=16;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so they are not independent.
  • B𝑃(𝐴∣𝐵)=13;𝑃(𝐴∣𝐵)=𝑃(𝐴), so they are independent.
  • C𝑃(𝐴∣𝐵)=118;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so they are not independent.
  • D𝑃(𝐴∣𝐵)=12;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so they are not independent.
  • E𝑃(𝐴∣𝐵)=16;𝑃(𝐴∣𝐵)=𝑃(𝐵), so they are independent.

Q18:

Mason and Liam are using their computers to take part in an online experiment on a website. When Mason presses the space bar on his keyboard, there is a 50% probability that his screen turns blue. When Liam presses the space bar on his keyboard, there is a 45% probability that his screen turns blue. If they both press their space bars, there is a 15% chance that both of their screens turn blue. Are “Mason’s screen turning blue” and “Liam’s screen turning blue” independent events?

  • ANo
  • BYes

Q19:

Suppose 𝑃(𝐴)=1942 and 𝑃(𝐵)=2942. The probability that neither event 𝐴 nor event 𝐵 occurs is 542. Calculate 𝑃(𝐴′∣𝐵′), and then evaluate whether events 𝐴 and 𝐵 are independent.

  • A𝑃(𝐴′∣𝐵′)=523;𝑃(𝐴′∣𝐵′)≠𝑃(𝐴′), so they are not independent.
  • B𝑃(𝐴′∣𝐵′)=2324;𝑃(𝐴′∣𝐵′)=𝑃(𝐴′), so they are independent.
  • C𝑃(𝐴′∣𝐵′)=513;𝑃(𝐴′∣𝐵′)≠𝑃(𝐴′), so they are not independent.
  • D𝑃(𝐴′∣𝐵′)=1942;𝑃(𝐴′∣𝐵′)=𝑃(𝐴′), so they are independent.
  • E𝑃(𝐴′∣𝐵′)=542;𝑃(𝐴′∣𝐵′)≠𝑃(𝐴′), so they are not independent.

Q20:

Consider the following Venn diagram.

Calculate the value of 𝑃(𝐵∣𝐴).

  • A12
  • B210
  • C25
  • D110
  • E310

Q21:

The figure shows a Venn diagram with some of the probabilities given for two events 𝐴 and 𝐵.

Work out 𝑃(𝐴∩𝐵).

Work out 𝑃(𝐵).

Work out 𝑃(𝐴∣𝐵).

Q22:

The figure shows a Venn diagram with some of the probabilities given for two events 𝐴 and 𝐵.

Work out 𝑃(𝐴∩𝐵).

Work out 𝑃(𝐴).

Work out 𝑃(𝐵∣𝐴).

  • A13
  • B12
  • C23
  • D34
  • E110

Q23:

For two events 𝐴 and 𝐵, 𝑃(𝐴)=0.4, 𝑃(𝐵)=0.7, and 𝑃(𝐴∪𝐵)=0.8.

Work out the value of 𝑟 in the Venn diagram.

Work out 𝑃(𝐵∣𝐴).

  • A34
  • B37
  • C14
  • D25
  • E310

Q24:

The given Venn diagram shows the probabilities of events 𝐴 and 𝐵 occurring or NOT occurring in different combinations.

Calculate 𝑃(𝐴) and 𝑃(𝐴∣𝐵), and then determine whether 𝐴 and 𝐵 are independent events.

  • A𝑃(𝐴)=1119 and 𝑃(𝐴∣𝐵)=1119;𝑃(𝐴∣𝐵)=𝑃(𝐴), so 𝐴 and 𝐵 are independent events.
  • B𝑃(𝐴)=319 and 𝑃(𝐴∣𝐵)=45;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so 𝐴 and 𝐵 are not independent events.
  • C𝑃(𝐴)=1119 and 𝑃(𝐴∣𝐵)=811;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so 𝐴 and 𝐵 are not independent events.
  • D𝑃(𝐴)=1119 and 𝑃(𝐴∣𝐵)=45;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so 𝐴 and 𝐵 are not independent events.
  • E𝑃(𝐴)=45 and 𝑃(𝐴∣𝐵)=45;𝑃(𝐴∣𝐵)≠𝑃(𝐴), so 𝐴 and 𝐵 are independent events.

Q25:

The given Venn diagram shows the probabilities of events 𝐴 and 𝐵 occurring or not occurring in different combinations.

Calculate the value of 𝑥.

  • A𝑥=0
  • B𝑥=449
  • C𝑥=1449
  • D𝑥=421
  • E𝑥=1421

Hence, calculate 𝑃(𝐴).

  • A𝑃(𝐴)=17
  • B𝑃(𝐴)=57
  • C𝑃(𝐴)=1721
  • D𝑃(𝐴)=521
  • E𝑃(𝐴)=13

Calculate 𝑃(𝐴∣𝐵).

  • A𝑃(𝐴∣𝐵)=14
  • B𝑃(𝐴∣𝐵)=34
  • C𝑃(𝐴∣𝐵)=45
  • D𝑃(𝐴∣𝐵)=0
  • E𝑃(𝐴∣𝐵)=47

Are 𝐴 and 𝐵 independent events?

  • AYes
  • BNo

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