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The two sides of a fair coin are labelled as $0$ to $1$. The coin is tossed two times independently. Let $M$ and $N$ denote the labels corresponding to the outcomes of those tosses. For a random variable $X$, defined as $X = \text{min}(M, N)$, the expected value $E(X)$ (rounded off to two decimal places) is ___________.

Concept:

If X be a random variable with a finite number of outcomes x1, x2, x3 .... occurring with probabilities p1, p2, p3....... respectively, the expectation of X is defined as:

$$E\left[ x \right] = \mathop \sum \limits_{i = 1}^k {x_i} \cdot {p_i} = {x_1}{p_1} + {x_2}{p_2} \ldots$$

Application:

It is given that the two sides of a fair coin are labelled as 0 and 1 and the coin is tossed two times independently.

∴ The total possible outcomes can be:

O = {(1,1), (1,0), (0,1), (0,0)}

The random variable X is defined as:

X = min (M, N)

So,

X1 = min {(1, 1)} = 1

X2 = min {(1, 0)} = 0

X3 = min {(0, 1)} = 0

X4 = min {(0, 0)} = 0

The probability of occurrence of ‘1’ will be:

$$P\left( 1 \right) = \frac{1}{4}$$

And the probability of occurrence of ‘0’ will be:

$$P\left( 0 \right) = \frac{3}{4}$$

We know that:

$$E\left[ x \right] = \mathop \sum \limits_{i = 1}^k {x_i} \cdot {P_i}$$

$$E\left[ x \right] = 0 \times \frac{3}{4} + 1 \times \frac{1}{4}$$

$$E\left[ x \right] = 0.25$$

150 points