Answer: 73.81%
Explanation:
We know that 5 out of 10 have broken screens. So each smartphone has a probability of 50% of having a broken screen.
The probability that at most 3 of them have a broken screen is:
The probability that none has a broken screen p = 0
This is 0 because we are selling six of them and only 5 of them have the screen in good condition.
The probabilities are calculated as:
P = A*6!/(6 - n)!*n!
where n is the number of broken phones.
the probability is equal to
if we have only one broken screen:
for the broken phone, we have a 5/10 probability.
for the other 5 nice phones, the probabilities will be 5/9. 4/8. 3/7, 2/6 and 1/5 (the number changes because when we select one of the phones, the total number of phones decreases)
A = (5*5*4*3*2*1)/(10*9*8*7*6*5)
and is similar thinking for 2 and 3 broken phones.
Probability that one has a broken screen:
P = ((5*4*3*2*1*5)/(10*9*8*7*6*5))*6 = 0.0238
Probability for two broken screen:
P = (5*4*3*2*5*4)/(10*9*8*7*6*5))*6*5/2 = 0.2381
For 3 broken phones the probability is:
P = ((5*4*3*5*4*3)/(10*9*8*7*6*5))*6*5*4/(2*3) = 0.4762
total probability = 0.0238 + 0.2381 + 0.4762 = 0.7381
or 73.81% in percentage form.