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Binomial variable counting ones in repeated die rolls

A standard die is rolled 6 times. Use a binomial variable to find the probability of rolling at least 4 ones.

Solution

(1) Labels.

Let .

Interpret: counts the ones appearing over 6 rolls.

We seek .

(2) Calculations.

Exclusive events:

Roll die until

Roll a fair die repeatedly. Find the probabilities that:

(a) At most 2 threes occur in the first 5 rolls.

(b) There is no three in the first 4 rolls, using a geometric variable.

Solution

(a)

(1) Label variables and events:

Use a variable to count the number of threes among the first six rolls.

Seek as the answer.

(2) Calculations:

Divide into exclusive events:

(b)

(1) Label variables and events:

Use a variable to give the roll number of the first time a three is rolled.

Seek as the answer.

(2) Compute:

Sum the PMF formula for :

(3) Recall geometric series formula:

For any geometric series:

Therefore:

Cubs winning the World Series

Suppose the Cubs are playing the Yankees for the World Series. The first team to 4 wins in 7 games wins the series. What is the probability that the Cubs win the series?

Assume that for any given game the probability of the Cubs winning is and losing is .

Solution

Method (a): We solve the problem using a binomial distribution.

(1) Label variables and events:

Use a variable . This counts the number of wins over 7 games. Thus, for example, is the probability that the Cubs win exactly 4 games over 7 played.

Seek as the answer.

(2) Calculate using binomial PMF:

Insert data:

Compute:

Convert :

Method (b): We solve the problem using a Pascal distribution instead.

(1) Label variables and events:

Use a variable . This measures the discrete wait time until the win. Thus, for example, is the probability that the Cubs win their game on game number .

Seek as the answer.

(2) Calculate using Pascal PMF:

Insert data:

Compute:

Convert :

Notice: The calculation seems very different than method (a), right up to the end!