A missile shoots down an enemy aircraft with a probability of 0.6 to find the number of missiles

A missile shoots down an enemy aircraft with a probability of 0.6 to find the number of missiles needed to destroy an enemy aircraft with a probability of 0.9.

1. The probability of the event Ai, consisting in the fact that with the i-th shot the missile will hit the target:

P (Ai) = 0.6;

2. Probability of the opposite event Ai ‘, that at the i-th shot the missile will miss:

P (Ai ‘) = 1 – 0.6 = 0.4.

3. Events Ai, like Ai ‘, are independent, therefore, the probability of event Xi that the missile will miss with i shots is:

P (Xi) = 0.4 ^ i,

and the probability of the opposite event Xi ‘that at least one of i missiles will destroy a missile is:

P (Xi ‘) = 1 – P (Xi) = 1 – 0.4 ^ n.

4. Find several values ​​of P (Xi ‘):

P (X1 ‘) = 1 – 0.4 = 0.6;
P (X2 ‘) = 1 – 0.4 ^ 2 = 1 – 0.16 = 0.84;
P (X3 ‘) = 1 – 0.4 ^ 3 = 1 – 0.064 = 0.936.
From the results obtained, it follows that three missiles are sufficient to destroy an enemy aircraft with a probability of 0.9.

Answer: 3 rockets.