Worksheet: Conditional Probability: Bayes's Theorem

In this worksheet, we will practice computing probabilities using Bayes’ rule.

Q1:

Suppose that 𝐴 and 𝐡 are events with probabilities 𝑃(𝐴)=0.63 and 𝑃(𝐡)=0.77. Given that 𝑃(𝐡|𝐴)=0.88, find 𝑃(𝐴|𝐡).

Q2:

Suppose that 𝐴 and 𝐡 are events in a random experiment. Given that 𝑃(𝐴)=0.39 and 𝑃(𝐡|𝐴)=0.88, find π‘ƒο€Ίπ΅βˆ£π΄ο†.

Q3:

Matthew rolls two fair dice numbered from one to six and records the results. Let 𝐴 be the event of rolling two numbers whose product is a square number and let 𝐡 be the event of rolling two numbers that are even.

Determine the probability of 𝐴.

  • A29
  • B14
  • C12
  • D518
  • E16

Determine the probability of 𝐡.

  • A29
  • B12
  • C14
  • D518
  • E16

Determine the probability of (𝐴∣𝐡).

  • A13
  • B29
  • C118
  • D112
  • E38

Determine the probability of (𝐡∣𝐴).

  • A112
  • B38
  • C13
  • D14
  • E118

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

  • Ano
  • Byes

Q4:

Suppose that 𝐴 and 𝐡 are events with probabilities 𝑃(𝐴)=0.45 and 𝑃(𝐡)=0.7. Given that 𝑃(𝐡|𝐴)=0.84, find 𝑃(𝐴|𝐡).

Q5:

Suppose that 𝐴 and 𝐡 are events with probabilities 𝑃(𝐴)=0.4 and 𝑃(𝐡)=0.6. Given that 𝑃(𝐡|𝐴)=0.72, find 𝑃(𝐴|𝐡).

Q6:

Suppose that 𝐴 and 𝐡 are events in a random experiment. Given that 𝑃(𝐴)=0.36 and 𝑃(𝐡|𝐴)=0.75, find π‘ƒο€Ίπ΅βˆ£π΄ο†.

Q7:

Suppose that 𝐴 and 𝐡 are events in a random experiment. Given that 𝑃(𝐴)=0.45 and 𝑃(𝐡|𝐴)=0.66, find π‘ƒο€Ίπ΅βˆ£π΄ο†.

Q8:

True or False: If we know the probabilities of event 𝐴 and event 𝐡 occurring and we know to probability of 𝐡 given 𝐴, Bayes’ rule tells us how to find 𝑃(𝐴∣𝐡).

  • AFalse
  • BTrue

Q9:

A professor awarded a final grade of A to 20% of the students. Of those who obtained a final grade of A, 70% obtained an A on the midterm exam. And of those students who failed to obtain a final grade of A, 10% earned an A on the midterm exam. Find the probability that a student with an A on the midterm exam will obtain a final grade of A.

  • A711
  • B750
  • C710
  • D110
  • E25

Q10:

Consider two boxes. Box A contains 2 red balls and 1 blue ball. Box B contains 3 blue balls and 1Β red ball. A coin is tossed. If it falls heads up, box A is selected and a ball is drawn. If it falls tails up, box B is selected and a ball is drawn. If the selected ball is red, find the probability that it is selected from box A.

  • A411
  • B23
  • C811
  • D12
  • E911

Q11:

Consider three bags. Bag A has 2white balls and 3 black balls. Bag B has 4 white balls and 1 black ball. Bag C has 3 white balls and 4 black balls. A bag is selected at random, and a ball drawn at random is found to be white.

Find the probability that bag A was selected.

  • A215
  • B1557
  • C13
  • D1457
  • E25

Q12:

We want to select a person from two committee selection pools. Pool A consists of 5 men and 10 women, while pool B consists of consists of 5 men and 7 women. A fair die is rolled. If the number on the die is greater than 5, then the person is selected from pool A, and if the number on the die is less than or equal to 5, then the person is selected from pool B. If you know that the person is a man, what is the probability that he is selected from pool A?

  • A56
  • B118
  • C16
  • D529
  • E429

Q13:

Suppose that P(𝐴)=0.4, P(𝐴∣𝐡)=0.5, and P(𝐡∣𝐴)=0.2. Find P(𝐡).

Q14:

Suppose that 𝐴 and 𝐡 are events with probabilities P(𝐴)=0.4 and P(𝐡)=0.64. Given that P(𝐴∣𝐡)=0.55, find P(𝐡∣𝐴).

Q15:

In a group of 96 people, 34 out of the 71 women have a smartphone, and 18 men do not have a smartphone. Determine the probability that a randomly picked smartphone owner in this group will be female.

  • A725
  • B3441
  • C7196
  • D2596
  • E3771

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