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1

WB JEE 2008

Subjective

If an unbiased coin is tossed n times. Find the probability that head appears an odd number of times.

Answer

.

Explanation

Let P = Probability of getting a head in a single trial = $${1 \over 2}$$.

Number of trials = n

and x = number of heads in n trials.

We have $$P(x = r) = {}^n{C_r}{p^r}{q^{n - r}}$$

$$ = {}^n{C_r}{\left( {{1 \over 2}} \right)^r}{\left( {{1 \over 2}} \right)^{n - r}} = {}^n{C_r}{\left( {{1 \over 2}} \right)^n}$$

Now,

P(x = odd) = $$P(x = 1) + P(x = 3) + (x = 5) + ...$$

$$ = {}^n{C_1}{\left( {{1 \over 2}} \right)^n} + {}^n{C_3}{\left( {{1 \over 2}} \right)^n} + {}^n{C_5}{\left( {{1 \over 2}} \right)^n} + ...$$

$$ = ({}^n{C_1} + {}^n{C_3} + {}^n{C_5} + ...){\left( {{1 \over 2}} \right)^n}$$

$$ = {2^{n - 1}} \times {1 \over {{2^n}}} = {1 \over 2}$$

$$\therefore$$ P(head appears an odd number of times) $$ = {1 \over 2}$$.

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