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Chapter 6: Q. 99 (page 406)

Roulette Marti decides to keep placing a 1bet on number 15in consecutive spins of a roulette wheel until she wins. On any spin, there's a 1-in- 38chance that the ball will land in the 15slot.

(a) How many spins do you expect it to take until Marti wins? Justify your answer.

(b) Would you be surprised if Marti won in 3or fewer spins? Compute an appropriate probability to support your answer.

Short Answer

Expert verified

a. The spins it take until Marti wins is 38.

b. No, its not surprised if Marti won in 3or fewer spins.

Step by step solution

01

Part (a) Step 1: Given Information

Number of consecutive spins=15

Number of chances =1in38

Number of slots=15

02

Part (a) Step 2: Explanation 

Given:

p=138

Probability (or mean) of a geometric variable is the reciprocal of the expected number:

=11/38

Hence, the number of spin will be38.

03

Part (b) Step 1: Given Information

Number of consecutive spins=15

Number of chances =1in 38

Number of slots=15

04

Part (b) Step 2: Explanation

Given:

p=138

Geometric probability formulae:

P(X=k)=qk-1p=(1-p)k-1p

Add the corresponding probabilities:

P(X≤3)=P(X=1)+P(X=2)+P(X=3)

≈0.0769

=7.69%

Because the probability is over 5%, it is not uncommon to win a prize in 3or fewer spins, so the result is not unexpected.

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Most popular questions from this chapter

A large auto dealership keeps track of sales and leases agreements made during each hour of the day. Let X= the number of cars sold and Y= the number of cars leased during the first hour of business on a randomly selected Friday. Based on previous records, the probability distributions of Xand Yare as follows:

Define D=X-Y.

Compute σDassuming that XandY are independent. Show your work

A deck of cards contains 52cards, of which 4are aces. You are offered the following wager: Draw one card at random from the deck. You win \(10if the card drawn is an ace. Otherwise, you lose \)1. If you make this wager very many times, what will be the mean amount you win?

(a) About −\(1, because you will lose most of the time.

(b) About \)9, because you win \(10but lose only \)1.

(c) About −\(0.15; that is, on average you lose about 15cents. (d) About \)0.77; that is, on average you win about 77cents.

(e) About $0, because the random draw gives you a fair bet.

A large auto dealership keeps track of sales made during each hour of the day. Let X=the number of cars sold during the first hour of business on a randomly selected Friday. Based on previous records, the probability distribution of Xis as follows:

Compute and interpret the mean of X.

46. Cereal A company's single-serving cereal boxes advertise 9.63 ounces of cereal. In fact, the amount of cereal X in a randomly selected box follows a Normal distribution with a mean of 9.70 ounces and a standard deviation of 0.03 ounces.
(a) LetY=the excess amount of cereal beyond what's advertised in a randomly selected box, measured in grams ( 1 ounce =28.35grams). Find the mean and standard deviation of Y.
(b) Find the probability of getting at least 3 grams more cereal than advertised.

90. Normal approximation To use a Normal distribution to approximate binomial probabilities, why do we require that both np and n(1−p) be at least 10?

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