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

(a) A popular brand of cereal puts a card with one of five famous NASCAR drivers in each box. There is a $1/5$chance that any particular driver's card ends up in any box of cereal. Buy boxes of the cereal until you have all 5 drivers' cards.

(b) In a game of 4-Spot Keno, Lola picks 4 numbers from 1 to 80. The casino randomly selects 20 winning numbers from 1 to 80. Lola wins money if she picks 2 or more of the winning numbers. The probability that this happens is $0.259$. Lola decides to keep playing games of 4 -Spot Keno until she wins some money.

A Cereal brand puts a card with one of five famous NASCAR drivers in each box.

Chance probability$=\frac{1}{5}$

Number of drivers' cards $=5$

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 we count until the first success (we are interested in five successes), there is no geometric distribution.

Hence, it is not geometric.

Lola picks$=$4 numbers from 1 to 80

Winning numbers from 1 to 80 $=20$

Lola wins money if she picks $=2$or more of the winning numbers

The probability that this happens $=0.259$

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.

We measure the number of draws until the first success and$p=0.259$. Since the two outcomes of winning and not winning money are independent, a geometric distribution can be used.

Hence, it is geometric.