Question

Rockets are launched until the first successful launching
has taken place.if this does not occur within 5 attempts,the
experiment is halted and the equipment inspected.suppose that
there is a constant probability of 0.8 of having a successful
launching and that successive atempts are independent.Assume
that the cost of the first launching is K dollars while subsequent
launching cost K/3 dollars.whenever a successful launching take place,a certian amount of information
is obtained which may be expressed as financial gain of,say 'C' dollars.if 'T' is the net cost of this
experiment,find the probability distribution of T?

Answer 1

Solution.

We know that successive attempts are independent. In this case, use the formula for the geometric distribution, because geometric distribution is used for modeling number of failures until the first success.

We know that net cost is the total cost of the experiment minus any financial gain from it.

So after first successful attempt net cost

$$T_1 = K - C$$

After second successful attempt

$$T_2 = K + \frac{K}{3} - C = \frac{4K}{3} - C$$

Similarly,

$$T_3 = K + 2 \cdot \frac{K}{3} - C = \frac{5K}{3} - C$$ $$T_4 = 2K - C$$ $$T_5 = \frac{7K}{3} - C$$

In our problem

$$p = 0.8 \Rightarrow q = 1 - p = 0.2$$

Then

$$P(T = K - C) = 0.2^0 \cdot 0.8 = 0.8$$ $$P\left(T = \frac{4K}{3} - C\right) = 0.2^1 \cdot 0.8 = 0.16$$ $$P\left(T = \frac{5K}{3} - C\right) = 0.2^2 \cdot 0.8 = 0.032$$ $$P(T = 2K - C) = 0.2^3 \cdot 0.8 = 0.0064$$ $$P\left(T = \frac{7K}{3} - C\right) = 0.2^4 \cdot 0.8 = 0.00128$$

Answer:

$$P(T = K - C) = 0.8$$ $$P\left(T = \frac{4K}{3} - C\right) = 0.16$$ $$P\left(T = \frac{5K}{3} - C\right) = 0.032$$ $$P(T = 2K - C) = 0.0064$$ $$P\left(T = \frac{7K}{3} - C\right) = 0.00128$$