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Worksheet: Solving Conditional Probability Problems

Q1:

Suppose that 𝐴 and 𝐡 are two events. Given that 𝑃 ( 𝐴 ) = 0 . 5 2 and 𝑃 ( 𝐡 | 𝐴 ) = 0 . 7 5 , find 𝑃 ( 𝐴 ∩ 𝐡 ) .

Q2:

Suppose that 𝐴 and 𝐡 are two events. Given that 𝑃 ( 𝐴 ) = 0 . 8 and 𝑃 ( 𝐡 | 𝐴 ) = 0 . 8 5 , find 𝑃 ( 𝐴 ∩ 𝐡 ) .

Q3:

Suppose that 𝐴 and 𝐡 are two events. Given that 𝑃 ( 𝐴 ) = 0 . 7 5 and 𝑃 ( 𝐡 | 𝐴 ) = 0 . 8 8 , find 𝑃 ( 𝐴 ∩ 𝐡 ) .

Q4:

Suppose that 𝐴 and 𝐡 are events with probabilities 𝑃 ( 𝐴 ) = 0 . 3 4 and 𝑃 ( 𝐡 ) = 0 . 5 2 . Given that 𝑃 ( 𝐡 | 𝐴 ) = 0 . 6 1 5 , find 𝑃 ( 𝐴 βˆͺ 𝐡 ) .

Q5:

240 people are taking science classes. There are 104 people studying chemistry, 132 people studying biology, and 68 of those are studying both. What is the probability that a person is studying Chemistry given that they are studying Biology?

  • A 7 1 0
  • B 1 7 2 6
  • C 1 7 6 0
  • D 1 7 3 3

Q6:

Liam rolled a fair six-sided die. Determine the probability of him rolling a 6 or less given that he rolled an odd number.

  • A 1 2
  • B 1 6
  • C 5 6
  • D1

Q7:

For two events 𝐴 and 𝐡 , 𝑃 ( 𝐡 ) = 0 . 5 and 𝑃 ( 𝐴 ∣ 𝐡 ) = 0 . 3 . Determine the probability of 𝐴 ∩ 𝐡 .

Q8:

Consider the following Venn diagram.

Calculate the value of 𝑃 ( 𝐡 ∣ 𝐴 ) .

  • A 2 1 0
  • B 1 2
  • C 3 1 0
  • D 2 5
  • E 1 1 0

Q9:

Suppose 𝑃 ( 𝐡 ∣ 𝐴 ) = 1 2 and 𝑃 ( 𝐴 ) = 3 7 . What is the probability that events 𝐴 and 𝐡 both occur?

  • A 1 1 4
  • B 6 7
  • C 1 3 1 4
  • D 3 1 4
  • E 4 7

Q10:

For two events 𝐴 and 𝐡 , 𝑃 ( 𝐴 ) = 0 . 3 , 𝑃 ( 𝐡 ) = 0 . 4 , and 𝑃 ( 𝐴 ∩ 𝐡 ) = 0 . 2 .

Work out the probability of 𝐴 given 𝐡 .

  • A 2 3
  • B 2 2 5
  • C 3 2 5
  • D 1 2
  • E 3 5

Work out the probability of 𝐡 given 𝐴 .

  • A 2 3
  • B 3 2 5
  • C 3 5 0
  • D 1 2
  • E 3 5

Q11:

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

Calculate the value of π‘₯ .

  • A π‘₯ = 1 4 2 1
  • B π‘₯ = 0
  • C π‘₯ = 4 4 9
  • D π‘₯ = 4 2 1
  • E π‘₯ = 1 4 4 9

Hence, calculate 𝑃 ( 𝐴 ) .

  • A 𝑃 ( 𝐴 ) = 1 3
  • B 𝑃 ( 𝐴 ) = 5 7
  • C 𝑃 ( 𝐴 ) = 5 2 1
  • D 𝑃 ( 𝐴 ) = 1 7
  • E 𝑃 ( 𝐴 ) = 1 7 2 1

Calculate 𝑃 ( 𝐴 ∣ 𝐡 ) .

  • A 𝑃 ( 𝐴 ∣ 𝐡 ) = 3 4
  • B 𝑃 ( 𝐴 ∣ 𝐡 ) = 0
  • C 𝑃 ( 𝐴 ∣ 𝐡 ) = 1 4
  • D 𝑃 ( 𝐴 ∣ 𝐡 ) = 4 7
  • E 𝑃 ( 𝐴 ∣ 𝐡 ) = 4 5

Are 𝐴 and 𝐡 independent events?

  • Ayes
  • Bno

Q12:

Suppose 𝑃 ( 𝐴 ) = 2 5 and 𝑃 ( 𝐡 ) = 3 7 . The probability that event 𝐴 occurs and event 𝐡 also occurs is 1 5 . Calculate 𝑃 ( 𝐴 ∣ 𝐡 ) , and then evaluate whether events 𝐴 and 𝐡 are independent.

  • A 𝑃 ( 𝐴 ∣ 𝐡 ) = 2 5 ; 𝑃 ( 𝐴 ∣ 𝐡 ) = 𝑃 ( 𝐴 ) , so they are independent.
  • B 𝑃 ( 𝐴 ∣ 𝐡 ) = 1 5 ; 𝑃 ( 𝐴 ∣ 𝐡 ) β‰  𝑃 ( 𝐴 ) , so they are not independent.
  • C 𝑃 ( 𝐴 ∣ 𝐡 ) = 3 7 ; 𝑃 ( 𝐴 ∣ 𝐡 ) β‰  𝑃 ( 𝐴 ) , so they are not independent.
  • D 𝑃 ( 𝐴 ∣ 𝐡 ) = 7 1 5 ; 𝑃 ( 𝐴 ∣ 𝐡 ) β‰  𝑃 ( 𝐴 ) , so they are not independent.
  • E 𝑃 ( 𝐴 ∣ 𝐡 ) = 3 7 ; 𝑃 ( 𝐴 ∣ 𝐡 ) = 𝑃 ( 𝐡 ) , so they are independent.

Q13:

A bag contains three red marbles, two yellow marbles, and six blue marbles. You take a marble at random from the bag and record its color. Then, without replacing the first marble, you take a second marble from the bag and record its color.

Given that the first marble you take is red, what is the probability that the second marble you take is also red?

  • A 3 1 1
  • B 2 1 1
  • C 3 5 5
  • D 1 5
  • E 2 6 5 5

What is the probability that the second marble you take is red, regardless of the color of the first marble you take?

  • A 3 1 1
  • B 3 5 5
  • C 6 5 5
  • D 3 1 0
  • E 1 5

What is the probability that you take at least one red marble?

  • A 1 5
  • B 3 1 1
  • C 2 7 5 5
  • D 1 3
  • E 3 5 5

Q14:

Suppose that and are events in a random experiment. Given that and , find .

Q15:

Suppose 𝐴 and 𝐡 are two events. Given that 𝑃 ( 𝐴 ∩ 𝐡 ) = 2 3 and 𝑃 ( 𝐴 ) = 9 1 3 , find 𝑃 ( 𝐡 | 𝐴 ) .

  • A 9 1 3
  • B 2 3
  • C 1 3 9
  • D 2 6 2 7

Q16:

For two events and , , and . Find .

Q17:

Liam spins two spinners. The first has six equal sectors numbered from 1 to 6, and the second has four equal sectors numbered 1 to 4. He draws a two-way table to represent the sample space, as shown in the figure.

Work out the probability that at least one of the spinners lands on a 2.

  • A 1 4
  • B 5 1 2
  • C 1 6
  • D 3 8
  • E 1 5

Work out the probability that the sum of the numbers is even.

  • A 1 2
  • B 5 2 4
  • C 1 1 2 4
  • D 1 4
  • E 2 3

Work out the probability that at least one of the spinners lands on a 2 and the sum of the numbers is even.

  • A 1 4
  • B 3 1 6
  • C 1 6
  • D 7 8
  • E 1 8

Work out the probability that the sum of the numbers is even given that at least one of the spinners lands on a 2.

  • A 4 9
  • B 2 5
  • C 1 8
  • D 1 6
  • E 1 3

Q18:

In an experiment, Ethan is going to spin a fair three-sided spinner and a fair four-sided spinner. He draws a two-way table to show all of the possible outcomes.

1 2 3 4
1 (1, 1) (1, 2) (1, 3) (1, 4)
2 (2, 1) (2, 2) (2, 3) (2, 4)
3 (3, 1) (3, 2) (3, 3) (3, 4)

In his experiment, he wants to look at two events: spinning two numbers whose sum is prime, 𝐴 , and spinning at least one three, 𝐡 .

Find 𝑃 ( 𝐴 ) .

  • A 3 4
  • B 2 3
  • C 1 2
  • D 7 1 2
  • E 5 1 2

Find 𝑃 ( 𝐡 ) .

  • A 1 2
  • B 7 1 2
  • C 1 3
  • D 5 1 2
  • E 2 3

Find 𝑃 ( 𝐴 ∣ 𝐡 ) .

  • A 5 1 2
  • B 1 4
  • C 1 2
  • D 1 3
  • E 2 3

Find 𝑃 ( 𝐡 ∣ 𝐴 ) .

  • A 3 7
  • B 2 7
  • C 1 3
  • D 1 4
  • E 4 7

Find 𝑃 ( 𝐴 ∩ 𝐡 ) .

  • A 1 2
  • B 1 6
  • C 5 1 1 4
  • D 1 4
  • E 1 3

Is it true that 𝑃 ( 𝐴 ) 𝑃 ( 𝐡 ∣ 𝐴 ) = 𝑃 ( 𝐴 ∩ 𝐡 ) and 𝑃 ( 𝐡 ) 𝑃 ( 𝐴 ∣ 𝐡 ) = 𝑃 ( 𝐴 ∩ 𝐡 ) ?

  • Ayes
  • Bno

Q19:

In the final exam, 5 5 % of students failed chemistry, 2 5 % failed physics and 1 6 % failed both. What is the probability that a student passed physics given that they passed chemistry?

Q20:

Victoria either takes the bus to school or, if she misses it, she walks. The probability that she catches the bus on any given day is 0.4. If she catches the bus, the probability that she will get to school on time is 0.8, but if she misses the bus and has to walk, the probability of her being on time drops to 0.6.

Work out the probability that she catches the bus and is on time for school on a given day.

Work out the probability that she is on time for school on a given day, whether or not she catches the bus.

Hence, find the probability that she will be late for school on a given day.

Q21:

Anthony rolled two six-sided dice and added the two numbers.

Determine the probability of obtaining a score of 7.

  • A 5 3 6
  • B 1 3
  • C 7 3 6
  • D 1 6
  • E 1 5

Determine the probability of obtaining a score of 7 given that at least one three is rolled.

  • A 2 1 1
  • B 1 1 1
  • C 1 3
  • D 1 6
  • E 2 1 8

Q22:

For events 𝐴 and 𝐡 , where 𝑃 ( 𝐴 ∩ 𝐡 ) = 0 . 2 and 𝑃 ( 𝐴 ) = 0 . 5 , calculate 𝑃 ( 𝐡 ∣ 𝐴 ) .

Q23:

It has been found for two events 𝐴 and 𝐡 that 𝑃 ( 𝐴 ) = 0 . 7 , 𝑃 ( 𝐡 ) = 0 . 5 β€² , and 𝑃 ( 𝐴 βˆͺ 𝐡 ) = 0 . 9 .

Work out 𝑃 ( 𝐴 ∩ 𝐡 ) .

  • A 2 1 0
  • B 4 1 0
  • C 7 2 0
  • D 3 1 0
  • E 9 2 0

Work out 𝑃 ( 𝐴 ∣ 𝐡 ) .

  • A 3 5
  • B 2 5
  • C 4 5
  • D 3 7
  • E 7 1 0

Work out 𝑃 ( 𝐡 ∣ 𝐴 ) .

  • A 1 2
  • B 3 5
  • C 3 7
  • D 4 7
  • E 2 7

Q24:

For events 𝐴 and 𝐡 , where 𝑃 ( 𝐴 ∩ 𝐡 ) = 0 . 1 and 𝑃 ( 𝐡 ) = 0 . 2 , calculate the value of 𝑃 ( 𝐴 ∣ 𝐡 ) .

Q25:

For two events 𝐴 and 𝐡 , 𝑃 ( 𝐡 β€² ) = 0 . 3 and 𝑃 ( 𝐴 ∣ 𝐡 ) = 0 . 3 . Determine the probability of 𝐴 ∩ 𝐡 .