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Chapter 6: Q. 96 (page 405)

Geometric or not? Determine whether each of the following scenarios describes a geometric setting. If so, define an appropriate geometric random variable.

(a) Shuffle a standard deck of playing cards well. Then turn over one card at a time from the top of the deck until you get an ace.

(b) Lawrence is learning to shoot a bow and arrow. On any shot, he has about a 10%chance of hitting the bull's-eye. Lawrence's instructor makes him keep shooting until he gets a bull's-eye.

Short Answer

Expert verified

a. No Geometric distribution

b. Geometric distribution

Step by step solution

01

Part (a) Step 1: Given Information 

Shuffle a standard deck of playing cards well.

Turn over one card at a time from the top of the deck until get an ace.

02

Part (a) Step 2: Explanation 

Upon drawing two numbers, the variable has a geometric distribution if:

  • Two possible outcomes can result from the variable.
  • All the results are independent.
  • The variables measure the number of draws needed to reach the first success.
  • Each drawing has the same chance of being successful.

As there is no replacement, there is no geometric distribution, as every draw is a continuation of the previous one.

Hence, it is not geometric.

03

Part (b) Step 1: Given Information

Lawrence chance of hitting the bulls-eye=10%

His instructor makes him keep shooting until he gets a bull's eye.

04

Part (b) Step 2: Explanation 

Upon drawing two numbers, the variable has a geometric distribution if:

  • Two possible outcomes can result from the variable.
  • All the results are independent.
  • The variables measure the number of draws needed to reach the first success.
  • Each drawing has the same chance of being successful.

Geometric distribution, because the two possible outcomes are bull's eye and no bull's eye, shots are independent, we measure the number of shots until the first success and p=10%=0.10

Hence, it is geometric.

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