A binomial experiment is conducted with 7

trials. Each trial has a probability of 0.75 for

success.

a) Which of these is a theoretical probability P(r)

for r trials

i. P(2) = 0.012 iv. P(5) = 0.015

ii. P(3) = 0.058 v. P(6) = 0.311

iii. P(4) = 0.005

b) What is the theoretical probability that at least

4 of the trials are successful, rounded to three

decimal places?

c) What is the theoretical probability that at most

2 of the trials are successful, rounded to three

decimal places?

To calculate the theoretical probability here, we need three inputs: 1) the number of trials (which here is 7); the probability of success (which here is 0.75); and an integer representing the particular outcome (which here would be r: {0, 1, 2, 3, 4, 5, 6, 7}.

(a) Most of today's calculators have probability distribution functions built in. I've used binompdf(n,p,r) here. i. P(2) = 0.012 is correct; it's the result of typing in binompdf(7,0.75,2). ii. P(3) = 0.058 is correct. iii. P(4) = 0.005 is false; this probability is 0.173. iv. P(5) = 0.015 is false. v. P(6) = 0.311 is correct.

(b) The probability that at least four trials are successful is equivalent to P(4) + P(5) + P(6) + P(7). Another way in which to calculate this would be to add up P(0) + P(1) + P(2) + P(3) and subtract the resulting sum from 1.00: That comes to: 1 - (0.000 + 0.001 + 0.012 + 0.058), or 1 - 0.071, or 0.929