Name: Random Variables in Probability Theory
Uploaded: 2019-07-19
Duration: 8 min 57 s
Description: What do we mean by random variables? Watch this video to learn about random variables alongside continuous random variables and discrete random variables wit...

Question

A box contains $2$ green apples and $2$ red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable $X$ is the number of apples which have been taken when the process stops.

Draw up the probability distribution table for $X .$ Another box contains $2$ yellow peppers and $5$ orange peppers. Three peppers are taken from the box without replacement.

Accepted Answer

Given that,

Total Apple $= 2 Green + 2 Red = 4$

Let, X be the random variable, i.e. the number of Apple which have been taken when process is stop.

$∴ P (x = 2) = \frac{2}{4} \times \frac{1}{3}$

$∴ P (x = 2) = \frac{1}{6}$

out of $4, 2 red$ , one drawn then $3$ remaining of which $1$ is red

$\Rightarrow p (x = 3) = 2 \times (\frac{2}{4} \times \frac{2}{3} \times \frac{1}{2}) = \frac{1}{3}$

Out of 4, we don't know, 1st white or red that is why multiply by 2 and after same procedure, out of 4 white , 2 white Apple then remaining 3 2 red Apple and remaining 1 red Apple WRR or RWR

Multiply by 3 because of 3 possibility

$WWRR, WRWR,RWWR$

$\Rightarrow P (x = 3) = 2 \times (\frac{2}{4} \times \frac{2}{3} \times \frac{1}{2}) = \frac{1}{3}$

$\Rightarrow P (x = 4) = 3 \times (\frac{2}{4} \times \frac{1}{3} \times \frac{2}{2} \times \frac{1}{1}) = \frac{1}{2}$

Probability distribution of $X$

$x$ $2$ $3$ $4$

$P (X = x)$ $\frac{1}{6}$ $\frac{1}{3}$ $\frac{1}{2}$

Total pepper $= 2 yellow + 5 Orange = 7$

$\Rightarrow P (\frac{All three orange}{at least 2 from box orange}) = \frac{P (All three orange)}{P (at least 2 from box orange)}$

$\Rightarrow P (\frac{A}{B}) = \frac{P (A \cap B)}{P (B)}$

$= \frac{{}^{5}{C_{3}}}{{}^{7}{C_{3}}} \times \frac{{}^{7}{C_{3}}}{{}^{2}{C_{1}} \times {}^{5}{C_{2}} + {}^{5}{C_{3}}}$

$= \frac{10}{2 \times 10 + 10}$

$\Rightarrow P (\frac{A}{B}) = \frac{1}{3}$

Important Questions on Probability Distributions

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