The probability of a miss when fired is 0.3. Find the probability that out of 7 fired shots

The probability of a miss when fired is 0.3. Find the probability that out of 7 fired shots: a) they hit the target no more than three times; b) missed exactly 3 times; c) hit the target at least four times.

The probability of a miss q = 0.3;
The probability of hitting the target: p = 1 – q = 1 – 0.3 = 0.7.
We find the probabilities using the Bernoulli formula.
Pn (k) = C (n, k) p ^ k q ^ (n – k);
a) Probability of not hitting the target even once:
P7 (0) = C (7.0) 0.7 ^ 0 0.3 ^ 7 = 1 1 0.0002 = 0.0002.
Probability of hitting the target 1 time:
P7 (1) = C (7.1) 0.7 ^ 1 0.3 ^ 6 = 7 0.7 0.0017 = 0.0036.
The probability of hitting the target 2 times:
P7 (2) = C (7.2) 0.7 ^ 2 0.3 ^ 5 = 21 0.49 0.00243 = 0.025.
Probability of hitting the target 3 times:
P7 (3) = C (7.3) 0.7 ^ 3 0.3 ^ 4 = 35 0.343 0.0081 = 0.0972.
The probability that the target will be hit no more than three times:
P7 (≤3) = P7 (0) + P7 (1) + P7 (2) + P7 (3) = 0.0002 + 0.0036 + 0.025 + 0.0972 = 0.126.
b) The probability that they missed exactly 3 times, that is, they hit 4 times;
P7 (4) = C (7.4) 0.7 ^ 4 0.3 ^ 3 = 35 0.2401 0.027 = 0.2269.
c) The probability that they hit the target at least four times is the probability of the opposite of the event in point a) such that they miss at least three times.
P7 (> 4) = 1 – P7 (≤3) = 1 – 0.126 = 0.874.
Answer: a) 0.126; b) 0.2269; c) 0.874.