Question

you are dealt one card from a 52 - card deck. find the probability that you are dealt a heart or a 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 number of hearts

A standard deck has 13 hearts.

Step2: Calculate number of picture - cards

There are 3 picture - cards (Jack, Queen, King) in each of 4 suits, so 3×4 = 12 picture - cards.

Step3: Calculate number of heart picture - cards

There are 3 heart picture - cards (Jack, Queen, King of hearts).

Step4: Use the inclusion - exclusion principle

The number of cards that are either a heart or a picture card is \(n(\text{heart}\cup\text{picture})=n(\text{heart})+n(\text{picture}) - n(\text{heart}\cap\text{picture})\). Substituting the values, we get \(13 + 12-3=22\).

Step5: Calculate the probability

The probability \(P=\frac{n(\text{heart}\cup\text{picture})}{n(\text{total})}\), and since \(n(\text{total}) = 52\), then \(P=\frac{22}{52}=\frac{11}{26}\).

Answer:

\(\frac{11}{26}\)