Question

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. Billy likes to play cornhole in his free time. On any toss, he has about a $20\%$chance of getting a bag into the hole. As a challenge one day, Billy decides to keep tossing bags until he gets one in the hole.

Short answer

(a)No geometric distribution, the reason is that every draw is not independent of the previous ones, although there is no replacement.

(b)interested in the number of trials until he gets a bag in the hole, which is the first success.

Step-by-step solution

Part (a) Step 1: Given Information

whether of the following scenarios explain a geometric setting yes or not, If yes than define an appropriate geometric random variable.

Part (a) Step 2: Simplification

The geometric distribution of a random variable When a variable has two possible outcomes, each draw is independent of the previous ones. The variable computes the number of draws needed until the first success. Every draw has the same chance of winning.

There is no geometric distribution since each draw is not independent of the previous ones, even though there is no replacement.

Part (b) Step 1: Given Information

whether of the following scenarios explain a geometric setting yes or not, If yes than define an appropriate geometric random variable.

Part (b) Step 2: Simplification

A geometric setup has four conditions: independent trials, binary (success/failure), the chance of success is the same for each trial, and the variable of interest is the number of trials required to get the first success.

Binary: it is satisfied since Success=place bag in hole and Failure=do not place bag in hole. Independent trials: it is satisfied since each toss has an identical chance of landing a bag in the hole (as the probability is $20\%$)

Probability of success: it is satisfied, because there is a $20\%$possibility of getting a bag into the hold and therefore the probability of success is 20%.

It is satisfied since he is interested in the number of trials until he obtains a bag in the hole, which is the first success.

Despite the fact that all four prerequisites are met, the offered scenario explains a geometric situation.