Question

Let's assume you are having a party and have stocked your refrigerator with beverages. You have 12 bottles of Coors beer, 24 bottles of Rainier beer, 24 bottles of Schlitz light beer, 12 bottles of Hamms beer, 2 bottles of Heineken dark beer and 6 bottles of Pepsi soda. You go to the refrigerator to get beverages for your friends. In answering the following question assume you are randomly sampling without replacement. What is the probability your first three bottles selected are Pepsi's

Answer 1

0.000243 probability your first three bottles selected are Pepsi's

Explanation:

12 bottles of Coors beer, 24 bottles of Rainier beer, 24 bottles of Schlitz light beer, 12 bottles of Hamms beer, 2 bottles of Heineken dark beer and 6 bottles of Pepsi soda.

The number of Coors beer is 12

The number of Rainier beer is 24 bottles

The number of Schlitz light beer is 24 bottles

The number of Hamms beer is 12 bottles

The number of Heineken dark beer is 2 bottles

The number of Pepsi soda is 6 bottles

Therefore, total number of bottle is

12+24+24+12+2+6 = 80 bottles.

The probability of occurrence of an event A is calculated as follows

`P(A) = \frac{\text {number of favourable outcome}}{\text {total number of outcome}}`

The probability of selecting the first bottle of pepsi is

`\frac{\text {number of pepsi bottle}}{\text {number of bottle}} \\\\= (6)/(80)`

After first pick the total number of outcome is 79 and also the total number of pepsi bottle is reduced to 5

Therefore, the probability of slecting the second bottle of pepsi is p₂ = 5/79

After second pick the total number of outcome is 78 and also the total number of pepsi bottle is reduced to 4

Therefore, the probability of slecting the third bottle of pepsi is p₃ = 4/78

Hence, the probability of your first three bottles selected are Pepsi's is as follows

p₁ * p₂ * p₃ =

`=(6* 5 *4)/(80 * 79 * 78) \\\\= (1)/(4108) \\\\=0.000243`

0.000243 probability your first three bottles selected are Pepsi's

Answer 2

0.02% probability your first three bottles selected are Pepsi's

Explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

The order in which the bottles are selected is not important, so we use the combinations formula to solve this question.

Combinations formula:

`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

Desired outcomes:

Three bottles of Pepsi, from a set of 6. So

`D = C_(6,3) = (6!)/(3!(6-3)!) = 20`

Total outcomes:

12+24+24+12+2+6 = 80 bottles.

We choose 3.

`T = C_(80,3) = (80!)/(3!(80-3)!) = 82160`

What is the probability your first three bottles selected are Pepsi's

`p = (D)/(T) = (20)/(82160) = 0.0002`

0.02% probability your first three bottles selected are Pepsi's