68 % of kids watch tv daily. out of 12 kids what is the probability that:
a. Exactly six kids watch tv daily
b. Six or less kids watch tv daily
c. Seven or more kids watch tv daily
d. Find the mean for the distribution
This is a case of binomial distribution
Prob(watching) = .68/100 = 17/25
prob(not watching) = 8/25
a) Prob(exactly 6 out of 12 watching)
=C(12,6)(17/25)^6(8/25)^6 = ....
b) prob(6 or less)
= Prob(exactly 1) + Prob(exacly 2) + prob(exactly 3) + .. prob(exactly 6)
= C(12,1)(17/25)(8/25)^11 + C(12,2)(17/25)^2(8/25)^10 + .. + C(12,6)(17/25)^6(8/25)^6
do the others the same way
hope you have a good calculator.
We can use excel for this. do you know how I would set it up in excel?
To find the probability for each scenario, you need to use the binomial probability formula:
P(X=k) = nCk * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of exactly k successes
- n is the total number of trials (in this case, the number of kids)
- k is the number of desired successful outcomes (in this case, the number of kids watching TV daily)
- nCk (read as "n choose k") is the binomial coefficient, calculated as n! / (k! * (n-k)!), where ! denotes the factorial of a number
- p is the probability of success for each trial (in this case, the percentage of kids watching TV daily, which is 68% or 0.68)
- (1-p) is the probability of failure for each trial (1 - 0.68 = 0.32 in this case)
With that understanding, let's solve the given scenarios:
a. Exactly six kids watch TV daily:
Using the binomial probability formula, plug in n=12, k=6, and p=0.68:
P(X=6) = 12C6 * 0.68^6 * (1-0.68)^(12-6)
b. Six or fewer kids watch TV daily:
To find this probability, you need to calculate the probabilities of each scenario where k ranges from 0 to 6 (inclusive), and then sum them up:
P(X ≤ 6) = P(X=0) + P(X=1 ) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)
c. Seven or more kids watch TV daily:
Similarly, to find this probability, you need to calculate the probabilities of each scenario where k ranges from 7 to 12 (inclusive), and then sum them up:
P(X ≥ 7) = P(X=7) + P(X=8) + P(X=9) + P(X=10) + P(X=11) + P(X=12)
d. Find the mean for the distribution:
The mean (expected value) for a binomial distribution is given by the formula:
Mean = n * p
Plug in n=12 and p=0.68 to calculate the mean.