Question

you are dealt one card from a 52 - card deck. find the probability that you are dealt a heart or a non - picture card. click the icon to view a description of a standard deck of playing cards. the probability is (type an integer or a simplified fraction)

Explanation:

Step1: Calculate the number of hearts

There are 13 hearts in a 52 - card deck. So \(n(\text{hearts})=13\).

Step2: Calculate the number of non - picture cards

There are 12 picture cards (4 Jacks, 4 Queens, 4 Kings) in a 52 - card deck. So the number of non - picture cards \(n(\text{non - picture})=52 - 12=40\).

Step3: Calculate the number of hearts that are non - picture cards

There are 10 non - picture heart cards (Ace, 2 - 10 of hearts). Let \(A\) be the event of getting a heart and \(B\) be the event of getting a non - picture card. Then \(n(A\cap B) = 10\).

Step4: Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)

By the formula \(P(A)=\frac{n(A)}{n(S)}\), \(P(B)=\frac{n(B)}{n(S)}\) and \(P(A\cap B)=\frac{n(A\cap B)}{n(S)}\) where \(n(S) = 52\).
\(P(A)=\frac{13}{52}\), \(P(B)=\frac{40}{52}\) and \(P(A\cap B)=\frac{10}{52}\)
\(P(A\cup B)=\frac{13 + 40-10}{52}=\frac{43}{52}=\frac{7}{13}\)

Answer:

\(\frac{7}{13}\)