3. There are 12 red checkers and 3 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 7 times in 10 selections. Show your work. Help! I'm having problems. Please Explain it to me.
Answer:
P(success)=P(red)= 12/15= 0.8
n=7 r=5
10C7= 10!/(10-7)! = 10!/3! = 10*9*8*7*6*5*4*3!/ 3!= 10*9*8*7*6*5*4= 604,800
P(red7times)= 10C7 (0.8)^7(0.2)^3
= 604,800 (0.8)^7(0.2)^3
= 604,800 (0.2)(.008)
= 604,800(0.0016)
=967.68
To find the probability of selecting a red checker exactly 7 times in 10 selections, we can use the concept of probability.
First, let's define some variables:
- R: selecting a red checker
- B: selecting a black checker
Now, let's break down the problem into smaller steps:
Step 1: Determine the probability of selecting a red checker in a single selection.
Since there are 12 red checkers and 3 black checkers in the bag, the total number of checkers is 12 + 3 = 15.
Therefore, the probability of selecting a red checker is P(R) = 12/15 = 4/5.
Step 2: Determine the probability of selecting a black checker in a single selection.
Similarly, the probability of selecting a black checker is P(B) = 3/15 = 1/5.
Step 3: Determine the probability of selecting a red checker exactly 7 times in 10 selections.
To find this probability, we need to calculate the probability of selecting a red checker 7 times multiplied by the probability of selecting a black checker 3 times.
We can use the binomial probability formula for this calculation:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
In our case:
- n is the total number of selections (10)
- k is the number of successful outcomes (selecting a red checker) we are interested in (7)
- p is the probability of a successful outcome (selecting a red checker), which is 4/5
P(X=7) = (10C7) * (4/5)^7 * (1 - 4/5)^(10-7)
Using combinations, (10C7) = 10! / (7! * (10-7)!),
(10C7) = (10 * 9 * 8) / (3 * 2 * 1) = 120
P(X=7) = 120 * (4/5)^7 * (1/5)^3
Calculating this expression will give you the probability of selecting a red checker exactly 7 times in 10 selections.