Question

Find the probabilty of obtaining exactly two 2's in five rolls of a fair die.

Answer 1

`\rm{ (1250)/(7776)\; or\; 0.16075\\\\}`

Explanation:

This probability can be determined using combinatorics. The number of ways a sample of size r can be obtained from a larger set of n items is given by the formula

`C(n,r) = (n!)/(( r! (n - r)! ))`

We pronounce C(n, r) as n choose r

We are rolling the die five times. So n = 5

On each roll a 2 can appear or not

We need two 2's . This is our r

The number of ways two 2's can appear is given by 5 choose 2

`C(5,2)\\\\= (5!)/(( 2! (5 - 2)! ))\\\\= (5!)/(2! * 3! )\\\\`

5! is the factorial of 5, 2! is the factorial of 2 and 3! is the factorial of 3

5! = 5 x 3 x 3 x 2 x1

3! = 3 x 2 x 1

2! = 2 x1

`C(5,2) = (5* 4 * 3 * 2 * 1)/(((2 * 1) * (3 * 2 * 1))\\\\= 10\\`

You need not go through this laborious process to find 5 choose 2 there are calculators available

So two 2's can appear 10 times.

The probability of getting a 2 on one roll is 1/6 and the probability of not getting a 2 on a single roll is 1-1/6 = 5/6

This works out to a binomial probability which can be computed by the formula

P(x) = C(n, x) pˣqⁿ⁻ˣ

Where x is the number of times the desired outcome succeeds and n the total number of trials

Here x = 2, n = 5, C(5, 2) = 10

`\rm{So\:P(two\:2s\:on\:5\:throws)\: = 10 \cdot \left((1)/(6)\right)^2\cdot \left((5)/(6)\right)^3\\\\`

`{= \frac{10 \cdot 1\cdot 125}{{6^5}} = (1250)/(7776) = 0.16075\\\\`