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A coin is tossed five times. What is the probability that heads are observed more than three times?
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Verified Answer
The correct answer is:
$\frac{3}{16}$
Let P denote the probability of getting head in a single toss of a coin.
$\mathrm{p}=\frac{1}{2} \Rightarrow \mathrm{q}=\frac{1}{2}$
Let $X$ denote the no. of heads in 5 tosses of a coin. then, $\mathrm{X}$ is a binomial variate with parameters; $\mathrm{n}=5 \&$
$\mathrm{p}=\frac{1}{2}$.
$\therefore$ Req. probability $=\mathrm{P}(\mathrm{x}>3)$
$=1-\mathrm{P}(\mathrm{x} \leq 3)$
$=1-[\mathrm{P}(\mathrm{x}=0)+\mathrm{P}(\mathrm{x}=1)+\mathrm{P}(\mathrm{x}=2)+\mathrm{P}(\mathrm{x}=3)]$
$=1-\left[{ }^{5} \mathrm{C}_{0}+{ }^{5} \mathrm{C}_{1}+{ }^{5} \mathrm{C}_{2}+{ }^{5} \mathrm{C}_{3}\right] \frac{1}{2^{5}}$
$=1-[1+5+10+10] \frac{1}{32}$
$=\frac{32}{32}-\frac{26}{32}=\frac{6}{32}=\frac{3}{16}$
$\mathrm{p}=\frac{1}{2} \Rightarrow \mathrm{q}=\frac{1}{2}$
Let $X$ denote the no. of heads in 5 tosses of a coin. then, $\mathrm{X}$ is a binomial variate with parameters; $\mathrm{n}=5 \&$
$\mathrm{p}=\frac{1}{2}$.
$\therefore$ Req. probability $=\mathrm{P}(\mathrm{x}>3)$
$=1-\mathrm{P}(\mathrm{x} \leq 3)$
$=1-[\mathrm{P}(\mathrm{x}=0)+\mathrm{P}(\mathrm{x}=1)+\mathrm{P}(\mathrm{x}=2)+\mathrm{P}(\mathrm{x}=3)]$
$=1-\left[{ }^{5} \mathrm{C}_{0}+{ }^{5} \mathrm{C}_{1}+{ }^{5} \mathrm{C}_{2}+{ }^{5} \mathrm{C}_{3}\right] \frac{1}{2^{5}}$
$=1-[1+5+10+10] \frac{1}{32}$
$=\frac{32}{32}-\frac{26}{32}=\frac{6}{32}=\frac{3}{16}$
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