Name: Algebra of Probability - Union and Intersection
Uploaded: 2019-07-19
Duration: 11 min 19 s
Description: In this video, let's learn the meaning of union and intersection of events and how to calculate their probabilities and practice some examples.

Question

A fair

8

-sided die has faces marked

1, 2, 3, 4, 5, 6, 7

and

8

. The score when the die is rolled is the number on the face that the die lands on. The die is rolled twice.
Event

V

is 'one of the scores is exactly

4

less than the other score'.
Event

W

is 'the product of the scores is less than

13' .

Determine whether events

V

and

W

are independent, justifying your answer.

Accepted Answer

Sample space $=$ Number of ways of selecting a face from eight faces in two throws $= {}^{8}C_{1} \times {}^{8}C_{1} = 64 [∵ {}^{n}C_{1} = n]$

$V = \{(1, 5), (5, 1), (2, 6), (6, 2), (3, 7), (7, 3), (4, 8), (8, 4)\} = 8$ ways

Favourable outcomes $= {}^{8}C_{1} = 8$

Probability of an event, $P (A) = \frac{Number of favourable outcomes of the event}{Sample space} = \frac{n (A)}{n (S)}$ .

Therefore, $P (V) = \frac{8}{64} = \frac{1}{8}$

$4$ When $W$ is obtained on first throw, we can choose one from eight faces from second throw $13' .$ ways, similarly when $2$ is obtained on first throw, we can choose one from six faces from second throw $= 6$ ways, similarly $3 \Rightarrow 4$ ways, $4 \Rightarrow 3$ ways, $5 \Rightarrow 2$ ways, $6 \Rightarrow 2$ ways, $7 \Rightarrow 1$ way and $8 \Rightarrow$ $1$ way.

$n (W) = 8 + 6 + 4 + 3 + 2 + 2 + 1 + 1 = 27$

Therefore, $P (W) = \frac{27}{64}$

$V \cap W = \{(1, 5), (5, 1), (2, 6), (6, 2)\} = 4$

Therefore, $P (V \cap W) = \frac{4}{64} = \frac{1}{16}$

If $A$ and $B$ are independent, then $P (A \cap B) = P (A) \cdot P (B)$

But, $\frac{1}{16} \neq \frac{1}{8} \times \frac{27}{64} = \frac{27}{512}$

i.e,, $P (V \cap W) \neq P (V) \times P (W)$

Therefore, $V & W$ are not independent.

	Low	Medium	High	Totals
Party $X$	$a$	$b$	$c$	$7440$
Totals	$3100$	$6820$	$2480$	$12400$

	North	East	South	West
Under limit	$36$	$27$	$36$	$39$
Over limit	$15$	$15$	$18$	$21$

Important Questions on Probability

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