There are 12 red checkers and three black checkers in a bag. Checkers are selected one at a time, with replacement. each time the color of the checker is recorded, find the probability of selecting a red checker exactly seven times in 10 selections. Show your work
algebra 1
prob(red) = 12/15 = 4/5
prob(not red) = 1/5
prob( 7 reds of 10)
= C(10,7)(4/5)^7 (1/5)^3
= appr .2013
To find the probability of selecting a red checker exactly seven times in 10 selections, we need to consider the total number of possible outcomes and the number of favorable outcomes.
In this scenario, we have a bag with 12 red checkers and 3 black checkers. Each time a checker is selected, it is replaced back into the bag, which means the total number of checkers remains the same.
Now, let's break down the problem step by step:
Step 1: Determine the total number of possible outcomes.
For each selection, we have two possibilities: red or black. Therefore, the total number of possible outcomes for 10 selections is 2^10 since each selection is independent. Hence, there are a total of 1024 possible outcomes.
Step 2: Determine the number of favorable outcomes.
To have exactly seven red checkers selected, we need to choose red checkers in 7 out of the 10 selections. The number of ways to do this is determined by the binomial coefficient, which can be calculated using the formula:
C(n, k) = n! / (k! * (n - k)!)
where n is the total number of selections (10), and k is the number of successful outcomes (7) we want.
Using this formula, the number of favorable outcomes is calculated as follows:
C(10, 7) = 10! / (7! * (10 - 7)!) = 120
Step 3: Calculate the probability.
The probability of a favorable outcome is given by the ratio of the number of favorable outcomes to the number of possible outcomes:
P(favorable) = number of favorable outcomes / number of possible outcomes
In this case, the probability of selecting a red checker exactly seven times in 10 selections can be calculated as:
P(red exactly 7 times in 10 selections) = number of favorable outcomes / number of possible outcomes
P(red exactly 7 times in 10 selections) = 120 / 1024
P(red exactly 7 times in 10 selections) ≈ 0.1172
Therefore, the probability of selecting a red checker exactly seven times in 10 selections is approximately 0.1172, or 11.72%.