The members of a men’s club have a choice of wearing black or red vests to their club meetings. A study done over a period of many years determined that the percentage of black vests worn is 60%. If there are 10 men at a club meeting on a given night, what is the probability, to the nearest thousandth, that at least 8 of the vests worn will be black?
I know you're supposed to use combinations, but how do you write it all out?
BINOMIAL DISTRIBUTION, yes uses Combinations
n = 10
probability of black = .6
probability not black = .4
prob 8 = C(10,8) .6^8 .4^2
= 45*.6^8*.4^2 = .121
prob 9 = C(10,9) .6^9 .4^1
= 10 *.6^9 * .4 = .040
prob 10 = C(10,10) .6^10 .4^0
= 1 *.6^10 * 1 = .006
sum = .121+.040+.006 = .167
I got the Combinations from Pascal's triangle but you can use a table or calculate
C(n,k) = n!/[ k! (n-k)! ]
thanks!
To determine the probability that at least 8 of the vests worn will be black, you can use the concept of combinations in probability.
First, let's calculate the probability of exactly 8, 9, and 10 vests being black and then add those probabilities together.
The probability of a single vest being black is given as 60% or 0.6, and the probability of a single vest being red is 1 - 0.6 = 0.4.
To calculate the probability of exactly 8 vests being black, you will need to consider different combinations. Since there are 10 men, you need to choose 8 out of the 10 to wear black vests. This can be calculated using the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Where C(n, r) is the number of combinations of n items taken r at a time, n! denotes the factorial of n, and r! and (n-r)! are the factorials of r and (n-r) respectively.
So, for 8 black vests, the probability would be:
P(8 black vests) = C(10, 8) * (0.6)^8 * (0.4)^(10-8)
Similarly, for 9 black vests:
P(9 black vests) = C(10, 9) * (0.6)^9 * (0.4)^(10-9)
For 10 black vests:
P(10 black vests) = C(10, 10) * (0.6)^10 * (0.4)^(10-10)
To get the probability of at least 8 vests being black, you sum up these individual probabilities:
P(at least 8 black vests) = P(8 black vests) + P(9 black vests) + P(10 black vests)
Hope this helps clarify the concept!