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

Two ordinary fair dice are rolled. A score of $3$ points is awarded if exactly one die shows an odd number and there is also a difference of $1$ between the two numbers obtained. A player who rolls two even numbers is awarded a score of $2$ points, otherwise a player scores $1$ point.

Find the probability that a player scores $3$ points, given that the sum of the numbers on their two dice is greater than $9 .$

Accepted Answer

Two fair dies are rolled .

$1$ Total number of all favourable outcomes $2$ $\{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\}$

A score 3 awarded if exactly one die show an odd number and there is also a difference of 1 between two number.

A score 2 awarded if player rolled two even number otherwise player score 1 point.

Let, S be the number of scored obtained

$\Rightarrow P (s = 1) = \frac{17}{36} = \{(1, 1), (1, 3), (1, 4), (1, 5), (1, 6), (2, 5), (3, 3), (3, 5), (4, 1), (5, 1), (5, 3), (5, 5), (6, 6)\}$

$∴ P (S = 2) = \frac{9}{36} = \{(2, 2), (2, 4), (2, 6) (4, 2), (4, 4), (4, 6) (6, 2), (6, 4), (6, 6)\}$

$∴ P (S = 3) = \frac{10}{36} = \{(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4), (5, 6), (6, 5)\}$

$s$ $1$ $2$ $3$

$P (S = s)$ $\frac{17}{36}$ $\frac{9}{36}$ $\frac{10}{36}$

Probability $\Rightarrow P (3 Awarded when sum of number on two dice is greater than 9) = \frac{[P (s = 3) > 9]}{P (s = 3)}$

$= \frac{\{(6, 5) (5, 6)\}}{\{(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\}}$

$\Rightarrow \frac{[P (s = 3) > 9]}{P (s = 3)} = \frac{2}{6}$

$\Rightarrow \frac{[P (s = 3) > 9]}{P (s = 3)} = \frac{1}{3}$

$y$	$0$	$1$	$2$	$3$
$P (X = x)$	$0.10$	$0.12$	$0.36$	$0.42$

$y$	$0$	$1$	$2$	$3$	$4$
$P (Y = y)$	$0.03$	$2 p$	$0.32$	$p$	$0.05$

$y$	$0$	$1$	$2$	$3$	$4$
$P (Y = y)$	$0.03$	$2 p$	$0.32$	$p$	$0.05$

$y$	$1$	$3$	$9$	$m$
$P (V = v)$	$0.4$	$0.28$	$0.14$	$0.18$

Important Questions on Probability Distributions

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