A fair die is rolled four times. Calculate the probability of obtaining exactly two 5s. (Round your answer to four decimal places.)
A fair die is cast four times. Calculate the probability of obtaining at least two 6's. Round to the nearest tenth of a percent.
To calculate the probability of obtaining exactly two 5s when a fair die is rolled four times, we need to determine the total number of possible outcomes and the number of favorable outcomes.
The total number of possible outcomes when a fair die is rolled four times can be found using the formula for the number of outcomes in multiple events, which is the product of the number of outcomes for each event. Since there are six possible outcomes on a fair die (numbers 1 through 6), the total number of possible outcomes for rolling a fair die four times is 6^4 = 1296.
To find the number of favorable outcomes, we need to consider the different ways in which two 5s can occur in four rolls.
One possible arrangement of two 5s among the four rolls is: 5, 5, X, X, where X can be any number from 1 to 6. There are 6 options for each X, so the number of favorable outcomes for this arrangement is 6*6 = 36.
Now, we need to consider the different ways to arrange the two 5s out of the four rolls. Since there are four different positions in which the two 5s can occur, we need to multiply the number of favorable outcomes for each arrangement by the number of different ways to arrange the two 5s, which is given by the combination formula C(n, r) = n! / (r! * (n-r)!), where n is the total number of rolls (4) and r is the number of desired outcomes (2).
Using the combination formula, we can evaluate C(4, 2) = 4! / (2! * (4-2)!) = 4! / (2! * 2!) = 6.
Finally, we can calculate the number of favorable outcomes by multiplying the number of favorable outcomes for each arrangement by the number of different arrangements: 36 * 6 = 216.
Therefore, the number of favorable outcomes is 216.
The probability of obtaining exactly two 5s is given by the number of favorable outcomes divided by the total number of possible outcomes: 216/1296 = 0.1667.
Rounding to four decimal places, the probability of obtaining exactly two 5s when rolling a fair die four times is approximately 0.1667.
prob of getting a 5 = 4/36 = 1/9
prob not a 5 = 8/9
prob getting a 5 twice in 4 rolls
= C(4,2) (1/9)^2 (8/9)^2 = 6 (1/81)(64.81) = 128/2187 = appr .0585