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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

Condition of public school facilities. The National Center for Education Statistics (NCES) conducted a survey on the condition of America鈥檚 public school facilities. The survey revealed the following information. The probability that a public school building has inadequate plumbing is .25. Of the buildings with inadequate plumbing, the probability that the school has plans for repairing the building is .38. Find the probability that a public school building has inadequate plumbing and will be repaired.

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

A: {Sum of the numbers showing is odd.}

B: {Sum of the numbers showing is 9, 11, or 12.}

Are events A and B independent? Why?

For two independent events, A and B, P (A) = .4 and P(B) = .2 :

a. Find P (A鈭〣)

b. Find P (A/B)

c. Find P (A鈭�B)

Are you really being served red snapper? Red snapper is arare and expensive reef fish served at upscale restaurants. Federal law prohibits restaurants from serving a cheaper, look-alike variety of fish (e.g., vermillion snapper or lane snapper) to customers who order red snapper. Researchers

at the University of North Carolina used DNA analysis to examine fish specimens labeled 鈥渞ed snapper鈥� that were purchased from vendors across the country (Nature, July 15, 2004). The DNA tests revealed that 77% of the specimens were not red snapper but the cheaper, look-alike variety of fish.

a. Assuming the results of the DNA analysis are valid, what is the probability that you are actually served red snapper the next time you order it at a restaurant?

b. If there are five customers at a chain restaurant, all who have ordered red snapper, what is the probability that at least one customer is actually served red snapper?

Which events are independent?Use your intuitive understanding of independence to form an opinion about whether each of the following scenarios represents independent events.

a.The results of consecutive tosses of a coin.

b.The opinions of randomly selected individuals in a pre-election poll.

c.A Major League Baseball player's results in two consecutive at-bats.

d.The amount of gain or loss associated with investments in different stocks if these stocks are bought on the same day and sold on the same day 1 month later.

e.The amount of gain or loss associated with investments in different stocks bought and sold in different time periods, 5 years apart.

f.The prices bid by two different development firms in response to a building construction proposal.
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