Help me with this question please
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
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
(12/15)^7 * (3/15)^3 = ?
....but the probability of failure is not independent of the probability of success :)
In other words you did one of 120 sequences that yield 7 successes out of 10 trials.
To find the probability of selecting a red checker exactly 7 times in 10 selections, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
Since each checker is selected one at a time with replacement, there are 2 possible outcomes on each selection (red or black). As we have 10 selections, the total number of possible outcomes is 2^10 = 1024.
Number of favorable outcomes:
To select a red checker exactly 7 times in 10 selections, we have to calculate how many ways we can choose 7 red checkers from the 12 available and 3 black checkers from the 3 available.
We can use the combination formula, denoted as C(n, r) or nCr, to calculate the number of ways to choose r items from a set of n items. The formula is: C(n, r) = n! / (r!(n - r)!)
In this case, the number of ways to choose 7 red checkers from 12 available is C(12, 7) = 12! / (7!(12 - 7)!) = (12! / 7!) / 5! = 792.
Similarly, the number of ways to choose 3 black checkers from 3 available is C(3, 3) = 3! / (3!(3 - 3)!) = 1.
Therefore, the number of favorable outcomes is 792 * 1 = 792.
Now we can calculate the probability:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 792 / 1024
To simplify the fraction, we can cancel out the common factors:
Probability = (792 / 8) / (1024 / 8)
Probability = 99 / 128
So the probability of selecting a red checker exactly 7 times in 10 selections is 99 / 128.
binomial distribution
p(sucess) = 12/15 = 4/5 = .8
1-p = 1/5 = .2
n = 10
k = 7
P(x=7) = C(10,7) * .8^7 * .2^3
C(10,7) = 10!/[ 7! (3!) ]
= 10*9*8 / (3*2) = 80*9/6 = 120
so
P = (120)(.8^7)(.2^3) = 0.201