Examples

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) Labels.

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

Seek as the answer.

---

(2) Calculations.

Divide into exclusive events:

(b)

(1) Labels.

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

Seek as the answer.

---

Sum the PMF formula for .

(2) Compute:

---

(3)

Geometric series formula.

For any geometric series:

Apply formula:

Final answer is .

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

(a) Using a binomial distribution

(1) Label.

Let .

Thus is the probability that the Cubs win exactly 4 games over 7 played.

Seek as the answer.

---

(2) Calculate.

Use binomial PMF:

---

(3) Insert data:

---

(4) Compute:

Convert :

---

(b) Using a Pascal distribution

(1) Label.

Let .

Thus is the probability that the Cubs win their game on game number .

Seek as the answer.

---

(2) Calculate.

Use Pascal PMF:

---

(3) Insert data:

---

(4) Compute:

Convert :

Notice: The algebra seems very different, right up to the end!