Question

(a) A player tosses 3 fair coins. He wins Rs.500 if 3 heads appear, Rs.300 if 2 heads appear, Rs.100 if 1 head occurs. On the other hand, he loses Rs.1500 if 3 tails occur. Find the expected gain of the player and variance.
(b) The trouble shooting capability of an IC chip in a circuit is a random variable X whose distribution function is given by where x denote the number of years. Find the probability that the IC chip will work properly (i) less than 8 years (ii) beyond 8 years (iii) between 5 to 7 years

Answer 1

Answer on Question #45517 – Math - Statistics and Probability

Problem.

(a) A player tosses 3 fair coins. He wins Rs.500 if 3 heads appear, Rs.300 if 2 heads appear, Rs.100 if 1 head occurs. On the other hand, he loses Rs.1500 if 3 tails occur. Find the expected gain of the player and variance.

(b) The trouble shooting capability of an IC chip in a circuit is a random variable X whose distribution function is given by where x denote the number of years. Find the probability that the IC chip will work properly (i) less than 8 years (ii) beyond 8 years (iii) between 5 to 7 years

Solution.

(a) The probability that head appears equals $\frac{1}{2} = 0.5$ . The probability that tail appears equals $\frac{1}{2} = 0.5$ .

The probability that 3 heads appear equals $0.5^{3} = 0.125$ by Bernoulli trial.

The probability that 2 heads appear equals $\binom{3}{2} 0.5^2 0.5 = 0.375$ by Bernoulli trial.

The probability that 1 heads appear equals $\binom{3}{1} 0.5^2 0.5 = 0.375$ by Bernoulli trial.

The probability that 0 heads appear equals $\binom{3}{0} 0.5^2 0.5 = 0.125$ by Bernoulli trial.

Expected gain $E = 0.125 \cdot 500 + 0.375 \cdot 300 + 0.375 \cdot 100 + 0.125 \cdot (-1500) = 25 = E$ .

Variance $= 0.125 \cdot (500 - E)^{2} + 0.375 \cdot (300 - E)^{2} + 0.375 \cdot (100 - E)^{2} + 0.125 \cdot (-1500 - E)^{2} = 346953.125$ .

(b) The part of question is missed "whose distribution function is given by $f(x) = \ldots$ , where $x$ denote the number of years". Here $f(x)$ is the probability distribution function, $F(x)$ is the cumulative distribution function.

(i) The probability that the IC chip will work properly less than 8 years is $F(8)$ .

(ii) The probability that the IC chip will work properly beyond 8 years is $1 - F(8)$ .

(iii) The probability that the IC chip will work properly between 5 to 7 years 8 is $F(7) - F(5)$ .

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