.8 probability that a randomly selected adult will watch prime time tv live, instead of online, on DVR, etc.. Assume that seven adults are randomly selected. find the propbablity that fewer than three of the selected adults watch prime time live.
prob(watch live) = .8
prob(not live) = .2
fewer than three of 7 means: 0 live, 1 live, or 2 live
prob(of your stated event)
= C(7,0)(.8)^0 (.2)^7 + C(7,1)(.8)^1 (.2)^6 + C(7,2)(.8)^2 (.2)^5
= .0000128 + .0003584 + .0043008
= .004672
seems low since the distribution is skewed towards the other end
e.g. C(7,6)(.8)^6 (.2)^1 = .367 , that one entry alone accounts for over 1/3 of all the distribution.
this is a binomial probability ... prime or other ... p(P) = .8 ... p(O) = .2
(O + P)^8 = O^8 + 8 O^7 P + 28 O^6 P^2 + ... + P^8
sum the 1st three terms to find p(P<3)
... .2^8 + (8 * .2^7 * .8) + (28 * .2^6 * .8^2)
got the exponent wrong ... should be 7 , not 8
(O + P)^7 = O^7 + 7 O^6 P + 21 O^5 P^2 + ... + P^7
.2^7 + (7 * .2^6 * .8) + (21 * .2^5 * .8^2)
To find the probability that fewer than three of the selected adults watch prime time TV live, we need to calculate the probabilities for zero, one, and two adults watching live, and then add them together.
First, let's find the probability that zero adults watch prime time TV live. Since each adult has a 0.8 probability of not watching live, the probability that one selected adult does not watch live is 0.2. We raise this probability to the power of 7 (since we are selecting seven adults) to get:
P(X=0) = (0.2)^7 ≈ 0.000128
Next, let's find the probability that one adult watches prime time TV live while the others do not. We have seven possible scenarios. For each scenario, the probability of one adult watching live is 0.8, and the probability of the remaining six adults not watching is 0.2. So, we multiply these probabilities by the number of possible scenarios:
P(X=1) = 7 * (0.8)^1 * (0.2)^6 ≈ 0.0064
Lastly, let's find the probability that two adults watch prime time TV live while the others do not. The number of possible scenarios is calculated through the combination formula: C(7, 2) = 21. For each scenario, the probability of two adults watching live is (0.8)^2, while the probability of the remaining five adults not watching is (0.2)^5:
P(X=2) = 21 * (0.8)^2 * (0.2)^5 ≈ 0.028
To find the probability that fewer than three adults watch prime time TV live, we sum up the calculated probabilities:
P(X < 3) = P(X=0) + P(X=1) + P(X=2)
≈ 0.000128 + 0.0064 + 0.028
≈ 0.034528
Therefore, the probability that fewer than three of the selected adults watch prime time TV live is approximately 0.034528, or 3.45%.