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Chapter 3: Q3-56E (page 193)

Two fair coins are tossed, and the following events are defined:

A: [Observe one head and one tail.]

B: [Observe at least one head.]

a. Define the possible sample points and assign probabilities to each.

b. Draw a Venn diagram for the experiment, showing the sample points and events A and B.

c. Find P(A), P(B) andP(A鈭�B).

d. Use the formula for conditional probability to find P (A/B)and P (B/A). Verify your answer by inspecting the Venn diagram and using the concept of reduced sample spaces.

Short Answer

Expert verified

Answer

  1. P (A)= [(H, T), (T, H)],P (B)= [(H, H), (H, T), (T, H)]
  2. Fig.1 Venn diagram
  3. 0.5, 0.75 & 0.5
  4. 0.67, 1.

Step by step solution

01

Step-by-Step SolutionStep 1: Identify the sample points

Probability is the study of prior records or the quantity and kind of probable outcomes to predict an event's result.

Sample points =[(H, H), (H, T), (T, H), (T, T)]P (A)= [(H, T), (T, H)]P (B)= [(H, H), (H, T), (T, H)]

02

Draw a Venn diagram

Here are all the sample points and events of A and B.

03

Find the required probability

P (A) =24=12= 0.5

P (B) =34= 0.75

P (A鈭�B) =24=12= 0.5

Hence, the required probabilities are 0.5, 0.75 & 0.5.

04

Find the required probability

P (A/B) =P (A鈭�B)P (B)=0.500.75= 0.67

P (B/A) =P (A鈭�B)P (A)=0.500.50= 1

Hence, the required probabilities are 0.67, 1.

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Most popular questions from this chapter

A pair of fair dice is tossed. Define the following events:

A: [Exactly one of the dice shows a 1.]

B: [The sum of the numbers on the two dice is even.]

a. Identify the sample points in the events A,B,A鈭�B,A鈭�B,andAc.

b. Find the probabilities of all the events from part a by summing the probabilities of the appropriate sample points.

C. Using your result from part b, explain why A and B are not mutually exclusive.

d. Find P(A鈭�B) using the additive rule. Is your answer the same as in part b?

The sample space for an experiment contains five sample points with probabilities as shown in the table. Find the probability of each of the following events:

a. Either 1,2 or 3 occurs

b. Either 1,3 or 5 occurs

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Appeals of federal civil trials. The Journal of the American Law and Economics Association (Vol. 3, 2001) publishedthe results of a study of appeals of federal civil trials. Thefollowing table, extracted from the article, gives a breakdownof 2,143 civil cases that were appealed by either theplaintiff or the defendant. The outcome of the appeal, aswell as the type of trial (judge or jury), was determined foreach civil case. Suppose one of the 2,143 cases is selected

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c. Are A and B mutually exclusive events?

d. FindP(AC)

e. FindP(A鈭�B)

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