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Chapter 3: Q108SE (page 204)

Speeding linked to fatal car crashes. According to the National Highway Traffic and Safety Administration鈥檚 National Center for Statistics and Analysis (NCSA), 鈥淪peeding is one of the most prevalent factors contributing to fatal traffic crashes鈥� (NHTSA Technical Report, August 2005). The probability that speeding is a cause of a fatal crash is .3. Furthermore, the probability that speeding and missing a curve are causes of a fatal crash is .12. Given speeding is a cause of a fatal crash, what is the probability that the crash occurred on a curve?

Short Answer

Expert verified

The probability is 0.4.

Step by step solution

01

Important formula

The formula is P(A鈭�B)=P(A).P(B|A).

02

The probability that the crash occurred on a curve.

Here, P(G)=0.3=Speeding is a cause of a fatal crash.

P(F|G)=0.12=Speeding and missing a curve are causes of a fatal crash.

P(G鈭�F)=P(G)脳P(F|G)0.12=(0.38)P(F|G)P(F|G)=0.4

Therefore the probability is 0.4.

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Most popular questions from this chapter

Suppose the events B1and B2are mutually exclusive and complementary events, such thatP(B1)=.75andP(B2)=.25 Consider another event A such that role="math" localid="1658212959871" P(AB1)=.3, role="math" localid="1658213029408" P(AB2)=.5.

  1. FindP(B1鈭�A).
  2. FindP(B2鈭�A)
  3. Find P(A) using part a and b.
  4. Findrole="math" localid="1658213127512" P(B1A).
  5. Findrole="math" localid="1658213164846" P(B2A).

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

at random and both the outcome of the appeal and type of trial are observed.

Jury

Judge

Totals

Plaintiff trial win-reserved

194

71

265

Plaintiff trial win-affirmed/dismissed

429

240

669

Defendant trial win-reserved

111

68

179

Defendant trial win- affirmed/dismissed

731

678

1030

Total

1465

678

2143

a. Find P (A), where A = {jury trial}.

b. Find P (B), where B = {plaintiff trial win is reversed}.

c. Are A and B mutually exclusive events?

d. FindP(AC)

e. FindP(A鈭�B)

f. FindP(A鈭�B)

Most likely coin-tossing sequence. In Parade Magazine鈥檚 (November 26, 2000) column 鈥淎sk Marilyn,鈥� the following question was posed: 鈥淚 have just tossed a [balanced] coin 10 times, and I ask you to guess which of the following three sequences was the result. One (and only one) of the sequences is genuine.鈥�

(1) H HHHHHHHHH

(2) H H T T H T T H HH

(3) T TTTTTTTTT

  1. Demonstrate that prior to actually tossing the coins, thethree sequences are equally likely to occur.
  2. Find the probability that the 10 coin tosses result in all heads or all tails.
  3. Find the probability that the 10 coin tosses result in a mix of heads and tails.
  4. Marilyn鈥檚 answer to the question posed was 鈥淭hough the chances of the three specific sequences occurring randomly are equal . . . it鈥檚 reasonable for us to choose sequence (2) as the most likely genuine result.鈥� If you know that only one of the three sequences actually occurred, explain why Marilyn鈥檚 answer is correct. [Hint: Compare the probabilities in parts b and c.]

Chance of winning at 鈥渃raps.鈥� A version of the dice game鈥渃raps鈥� is played in the following manner. A player starts by rolling two balanced dice. If the roll (the sum of the two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or a 3 (called craps), the player loses. For any other roll outcome, the player continues to throw the dice until the original roll outcome recurs (in which case the player wins) or until a 7 occurs

(in which case the player loses).

a. What is the probability that a player wins the game on the first roll of the dice?

b. What is the probability that a player loses the game on the first roll of the dice?

c. If the player throws a total of 4 on the first roll, what is the probability that the game ends (win or lose) on the next roll?

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?
See all solutions

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