According to an airline, flights on a certain route are on time 80 % of the time. suppose 10 flights are randomly selected and the number of on-time flights is recorded. (a) explain why this is a binomial experiment. (b) find and interpret the probability that exactly 7 flights are on time. (c) find and interpret the probability that fewer than 7 flights are on time. (d) find and interpret the probability that at least 7 flights are on time. (e) find and interpret the probability that between 5 and 7 flights, inclusive, are on time.

(a) This is a binomial experiment since there are only two possible results for each data point: a flight is either on time (p = 80% = 0.8) or late (q = 1 - p = 1 - 0.8 = 0.2).
(b) Using the formula:
P(r out of n) = (nCr)(p^r)(q^(n-r)), where n = 10 flights, r = the number of flights that arrive on time:
P(7/10) = (10C7)(0.8)^7 (0.2)^(10 - 7) = 0.2013
Therefore, there is a 0.2013 chance that exactly 7 of 10 flights will arrive on time.
(c) Fewer than 7 flights are on time means that we must add up the probabilities for P(0/10) up to P(6/10).
Following the same formula (this can be done using a summation on a calculator, or using Excel, to make things faster):
P(0/10) + P(1/10) + ... + P(6/10) = 0.1209
This means that there is a 0.1209 chance that less than 7 flights will be on time.
(d) The probability that at least 7 flights are on time is the exact opposite of part (c), where less than 7 flights are on time. So instead of calculating each formula from scratch, we can simply subtract the answer in part (c) from 1.
1 - 0.1209 = 0.8791.
So there is a 0.8791 chance that at least 7 flights arrive on time.
(e) For this, we must add up P(5/10) + P(6/10) + P(7/10), which gives us
0.0264 + 0.0881 + 0.2013 = 0.3158, so the probability that between 5 to 7 flights arrive on time is 0.3158.

According to the flight on certain routes, the experiment is a binomial experiment and this will be determined by observing the given data.

Given :

Flights on a certain route are on-time 80% of the time.10 flights are randomly selected and the number of on-time flights is recorded.

a). There are only two possible results for each data point, a flight is either on time (p = 80% = 0.8) or late (q = 1 - p = 1 - 0.8 = 0.2). Therefore it is a binomial experiment.

b). The probability that exactly 7 flights are on time

c). Fewer than 7 flights are on time means:

The probability that fewer than 7 flights are on time = 0.1209

d) The probability that at least 7 flights are on time = 1 - 0.1209 = 0.8791

e) The probability that between 5 and 7 flights, inclusive, are on time.

= 0.3158

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