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

Three dice are thrown. If a $1$ or a $6$ is rolled somewhere on these three dice, you will be paid $$ 1$ , but if neither are rolled you will pay $$ 5$ . You play the game,

Complete the table shows the probability distribution of $X$ , 'the number of dollars won in a game'.

$x$	$P (X = x)$
$- 5$
$1$

Accepted Answer

Let $X$ be the event of getting paid $$ 1$ by getting $1$ or $6$ on the three dice somewhere.

Number of independent events, $X$

The probability of getting $1$ or $6$ in the six faced dice, $P (X = x)$

Using binomial distribution, $X ~ B (3, \frac{1}{3})$

If $X$ is a discrete random variable which is binomial distributed , $X ~ B (n, p)$ then the probability of obtaining $r$ successes out of $n$ independent trials, when $p$ is the probability of success for each trial, is $P (X = r) = (\begin{matrix} n \\ r \end{matrix}) p^{r} {(1 - p)}^{n - r}$

The probability that the player will lose, i.e player will pay $$ 5$ is $P (X = 0)$

$\Rightarrow P (X = 0) = (\begin{matrix} 3 \\ 0 \end{matrix}) {(\frac{1}{3})}^{0} {(1 - \frac{1}{3})}^{3 - 0}$

$\Rightarrow P (X = 0) = {}^{3}C_{0} {(\frac{1}{3})}^{0} {(1 - \frac{1}{3})}^{3}$

$\Rightarrow P (X = 0) = 1 \times 1 \times {(\frac{2}{3})}^{3}$

$\Rightarrow P (X = 0) = \frac{8}{27}$

$\Rightarrow P (X = 0) = \frac{8}{27}$ .....(i)

The probability that the player will win $$ 1$ is $P (X \geq 1)$

$\Rightarrow P (X \geq 1) = 1 - P (X = 0)$

$\Rightarrow P (X \geq 1) = 1 - \frac{8}{27}$ (from (i))

$\Rightarrow P (X \geq 1) = \frac{19}{27}$

The probability that the player will win $$ 1$ is $\frac{19}{27}$ .

Hence, the table is

$x$ $P (X = x)$

$- 5$ $\frac{8}{27}$

$1$ $\frac{19}{27}$

Important Questions on Valid Comparisons and Informed Decisions: Probability Distributions

Learn and Prepare for any exam you want !

Three dice are thrown. If a 1 or a 6 is rolled somewhere on these three dice, you will be paid $1, but if neither are rolled you will pay $5. You play the game, Complete the table shows the probability distribution of X, 'the number of dollars won in a game'. x P(X=x) -5 1

Important Questions on Valid Comparisons and Informed Decisions: Probability Distributions

Learn and Prepare for any exam you want !