The probability of hitting a target with one shot is 0.8. 7 shots were fired.

The probability of hitting a target with one shot is 0.8. 7 shots were fired. Find the probability that there was: a) four hitting the target; b) six defeats; c) no more than six defeats.

Let us introduce the notation:

n is the number of shots;

k is the number of successful shots;

p is the probability of hitting the target with the next shot;

q is the probability of a miss during the next shot.

By assumption, n = 7 and p = 0.8. Let’s calculate the value of q.

q = 1 – 0.8 = 0.2.

Item A.

We need Bernoulli’s formula:

Pn (k) = p ^ k * q ^ (n – k).

With this formula, we can calculate the probability that the target has been hit four times. In this case, k = 4.

P7 (4) = C47 * p ^ 4 * q ^ (7 – 4);

C47 = 7! / (4! * (7 – 4)!) = 7! / (4! * 3!) = (5 * 6 * 7) / (2 * 3) = 35;

P7 (4) = 35 * p ^ 4 * q ^ 3 = 35 * 0.8 ^ 4 * 0.2 ^ 3 = 35 * 0.4096 * 0.008 = 0.114688 ≈ 0.115 = 11.5%.

Item B.

We use Bernoulli’s formula again. Let’s calculate the probability that the target was hit six times. In this case, k = 6.

P7 (6) = C6 / 7 * p ^ 6 * q ^ (7 – 6);

C67 = 7! / (6! * (7 – 6)!) = 7! / (6! * 1!) = 7;

P7 (6) = 7 * p ^ 6 * q ^ 1 = 7 * 0.8 ^ 6 * 0.2 = 7 * 0.262144 * 0.2 = 0.3670016 ≈ 0.367 = 36.7%.

Item B.

Let’s say event G happens if the target is hit seven times. Let us calculate the probability of event G.

P (G) = P7 (7) = p ^ 7 = 0.8 ^ 7 = 0.2097152.

Event Ḡ will occur if the target is hit less than seven times (that is, no more than six times). Let us calculate the probability of the event Ḡ.

P (Ḡ) = 1 – P (G) = 1 – 0.2097152 = 0.7902848 ≈ 0.790 = 79.0%.

Answer:

a) approximately 11.5%;

b) approximately 36.7%;

c) approximately 79.0%.