Question

A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, “the number is even”, and B be the event, “the number is red”. Are A and B independent?

Answer 1

Hint: This problem can be solved using probability of an event. First we need to find the probability of the events by dividing the favorable outcomes with total outcomes. If two events are independent, then the incidence of one event does not affect the probability of the other event. This can be given by${\text{P}}\left( {{\text{A}} \cap {\text{B}}} \right) = {\text{P(A)}}{\text{.P(B)}}$

Complete step-by-step answer:
Given, a die marked 1, 2, 3 in red and 4, 5 and 6 in green is tossed.
Therefore, S = {1, 2, 3, 4, 5, 6}
Let the two events be
A: the number is even
B: the number is red
Probability of an event = $\dfrac{{{\text{favourable outcomes}}}}{{{\text{total outcomes}}}}$
Even Numbers on the die are 2, 4 and 6.
That is, A= {2, 4, 6}
Therefore, probability of getting even number from the die, ${\text{P(A)}} = \dfrac{3}{6} = \dfrac{1}{2}$
There are two colours of the die – Red and Green
That is, B= {1, 2, 3}
Probability of getting red colour from the die, ${\text{P}}({\text{E}}) = \dfrac{1}{2}$
${\text{P}}\left( {{\text{A}} \cap {\text{B}}} \right)$= Even number in red colour.
Only 1 is red and even.
Therefore, probability of getting red colour and even number$ = $${\text{P}}\left( {{\text{A}} \cap {\text{B}}} \right)$$ = \dfrac{1}{6}$
Now, two events A and B are independent if ${\text{P}}\left( {{\text{A}} \cap {\text{B}}} \right) = {\text{P(A)}}{\text{.P(B)}}$
${\text{P}}\left( {\text{A}} \right).{\text{P}}\left( {\text{B}} \right) = \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4} \ne \dfrac{1}{6} \ne {\text{P}}\left( {{\text{A}} \cap {\text{B}}} \right)$
That is, ${\text{P}}\left( {{\text{A}} \cap {\text{B}}} \right) \ne {\text{P(A)}}{\text{.P(B)}}$
Therefore, A and B are not independent

Note: Two independent events will be when we are rolling a dice and flipping a coin. On the other hand, a dependent event is one which gets affected by the result of a second event. Statistically, an event A is to be independent of another event B, if the conditional probability of A given B, i.e, ${\text{P(A}}|{\rm B})$is equal to the unconditional probability of A. ${\text{P(B)}} \ne {\text{0}}$.
${\text{P(A}}|{\rm B}) = {\text{P(A)}}$