Abinomial 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?

Step-by-step explanation:

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