Name: Addition Theorem of Probability: Illustrations
Uploaded: 2019-07-19
Duration: 3 min 27 s
Description: If A and B are two events associated with a random experiment, how do we find the probability of at least one of them? Let's watch and find out in this video.

Question

A coin is tossed 5-times. Find the mean and variance of the probability distribution of appearance of heads on the tosses.

Accepted Answer

Let $X$ denote the number of heads in the $5$ tosses of coin. Then, $X$ is a random variable that can take the values $0, 1, 2, 3, 4 or 5$ .

$P (X = r) = {}^{5}C_{r} {(\frac{1}{2})}^{5 - r} {(\frac{1}{2})}^{r} = {}^{5}C_{r} {(\frac{1}{2})}^{5}$

$P (X = 0) =$ Probability of getting no head $= {}^{5}C_{0} {(\frac{1}{2})}^{5} = \frac{1}{32}$

$P (X = 1) =$ Probability of getting one head $= {}^{5}C_{1} {(\frac{1}{2})}^{5} = \frac{5}{32}$

$P (X = 2) =$ Probability of getting two heads $= {}^{5}C_{2} {(\frac{1}{2})}^{5} = \frac{10}{32}$

$P (X = 3) =$ Probability of getting three heads $= {}^{5}C_{3} {(\frac{1}{2})}^{5} = \frac{10}{32}$

$P (X = 4) =$ Probability of getting four heads $= {}^{5}C_{4} {(\frac{1}{2})}^{5} = \frac{5}{32}$

$P (X = 5) =$ Probability of getting five heads $= {}^{5}C_{5} {(\frac{1}{2})}^{5} = \frac{1}{32}$

The probability distribution of $X$ is

$x_{i} = X$ $0$ $1$ $2$ $3$ $4$ $5$

$p_{i} = P (X)$ $\frac{1}{32}$ $\frac{5}{32}$ $\frac{10}{32}$ $\frac{10}{32}$ $\frac{5}{32}$ $\frac{1}{32}$

$p_{i} x_{i}$ $0$ $\frac{5}{32}$ $\frac{20}{32}$ $\frac{30}{32}$ $\frac{20}{32}$ $\frac{5}{32}$

$p_{i} {x_{i}}^{2}$ $0$ $\frac{5}{32}$ $\frac{40}{32}$ $\frac{90}{32}$ $\frac{80}{32}$ $\frac{25}{32}$

Mean, $E (X) = \sum p_{i} x_{i} = \frac{5 + 20 + 30 + 20 + 5}{32} = \frac{80}{32} = \frac{5}{2} = 2.5$

$Var (X) = \sum p_{i} {x_{i}}^{2} - {(\sum p_{i} x_{i})}^{2} = (\frac{5 + 40 + 90 + 80 + 25}{32}) - (\frac{25}{4})$

$\Rightarrow Var (X) = (\frac{5 + 40 + 90 + 80 + 25}{32}) - (\frac{25}{4}) = \frac{240}{32} - \frac{200}{32}$

$\Rightarrow Var (X) = \frac{40}{32} = \frac{5}{4} = 1.25$ .

A coin is tossed 5-times. Find the mean and variance of the probability distribution of appearance of heads on the tosses.

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