91影视

We are given information of cards and a two-way table summarizes the sample space for this chance process based on whether or not the card is a face card and whether or not the card is a heart.

We need to find out are the events 鈥渉eart鈥� and 鈥渇ace card鈥� independent .

Two events are independent if probability of one event does not affect probability of another event.

Probability of face card in deck of card=$P$(Face card)=role="math" localid="1653922848957" $\frac{No.offavourableoutcomes}{No.ofpossibleoutcomes}=\frac{\cancel{12}}{\cancel{52}}=\frac{3}{13}$

Probability of heart in deck of card= P(heart)=$\frac{No.offavourableoutcome}{no.oftotaloutcome}=\frac{13}{52}$

Probability of face card and heart=P(face card and heart)=$\frac{3}{52}$

According to definition of conditional probability:

P(Face card | Heart)=$\frac{P(Facecardandheart)}{P\left(Heart\right)}=\frac{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$52$}\right.}}{{\displaystyle \raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$52$}\right.}}=\frac{3}{13}$

For event to be independent $P(A|B)=P(A)$

From above probabilities calculated by us we see that : P(Face card)= P(Face card | Heart) = $\frac{3}{13}$

Hence, events 鈥渉eart鈥� and 鈥渇ace card鈥� independent.