During a clothing sale, 1/4 of the store merchandise is reduced in price. Find the probablility that 3 of 5 randomly-selectled shirts have reduced prices.
my guess is to multiply 1/4 and 3/5 but that has to be wrong
To find the probability that 3 out of 5 randomly-selected shirts have reduced prices during the clothing sale, you need to use the concept of combinations.
First, let's calculate the total number of ways to select 5 shirts from the store merchandise. This can be done using combinations, denoted as "C(n, r)" which calculates the number of ways to select "r" items from a set of "n" items without regard to order.
In this case, we want to find C(n=5, r=5), which is equal to 1, since there is only one way to select all 5 shirts from a set of 5.
Next, let's calculate the number of ways to select 3 shirts with reduced prices from the 1/4 of the store merchandise that is reduced. Since we are choosing 3 shirts out of this reduced price selection, we can use C(n=1/4 * total_number_of_shirts, r=3).
C(n=1/4 * 5, r=3) = C(n=1, r=3) = (1!)/(3! * (1-3)!) = (1)/(3! * (-2)!) = 1 / (3 * 2 * 1) = 1 / 6 = 1/6
Now, let's calculate the number of ways to select the other 2 shirts from the remaining 3/4 of the store merchandise that is not reduced. Since we are choosing 2 shirts from this selection, we can again use the combinations formula:
C(n=3/4 * total_number_of_shirts, r=2) = C(n=3/4 * 5, r=2) = C(n=15/4, r=2) = (15/4!)/(2! * (15/4-2)!) = (15/4 * 3/4)/(2! * (15/4-2)!) = (45/16)/(2 * (7/4)!) = (45/16)/(2 * (7/4 * 3/4)!) = (45/16)/(2 * (7/12)!) = (45/16)/(2 * (7/12)) = (45/16)/(14/12) = (45/16)/(7/6) = (45/16) * (6/7) = 270/112 = 135/56 = 135/8 * 1/7 = 135/56
Finally, we can calculate the probability by dividing the number of favorable outcomes (number of ways to select 3 shirts with reduced prices and 2 shirts without reduced prices) by the total number of possible outcomes (number of ways to select any 5 shirts):
P(3 out of 5 shirts have reduced prices) = (number of favorable outcomes) / (total number of possible outcomes)
= (1/6 * 135/56) / (1/1)
= (1/6 * 135/56)
= 135/336
= 45/112
Therefore, the probability that 3 out of 5 randomly-selected shirts have reduced prices during the clothing sale is 45/112.