Question

in a game show, the two of the game show contestants pick a single card from a standard deck and payoffs are based on the card chosen. express your answers as reduced fractions.

find the probability that your card is

part 1 of 8
(a) the 4 of clubs.
the probability that the first card picked up was a 4 of clubs is

part 2 of 8
(b) a black card.
the probability that the first card picked up was a black card is

part 3 of 8
(c) a queen.
the probability that the first card picked up was a queen is

part 4 of 8
(d) a black 10.
the probability that the first card picked up was a black 10 is

part 5 of 8
(e) a red card or a 3.
the probability that the first card picked up was a red card or a 3 is

part 6 of 8
(f) a club and a 4.
the probability that the first card picked up was a club and a 4 is

part 7 of 8
(g) a 2 or an ace.
the probability that the first card picked up was a 2 or an ace is

Explanation:

Step1: Use the formula for probability

The formula for probability is \(P(A)=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of elements in event \(A\) and \(n(S)\) is the number of elements in the sample space. A standard deck has \(n(S) = 52\) cards.

Step2: Calculate \(n(A)\) for a 2 or an ace

There are 4 cards with the number 2 (one for each suit: hearts, diamonds, clubs, spades) and 4 cards with the number ace (one for each suit). But we need to make sure there is no double - counting. Since a card cannot be both a 2 and an ace at the same time, \(n(A)=4 + 4=8\).

Step3: Calculate the probability

Using the formula \(P(A)=\frac{n(A)}{n(S)}\), we substitute \(n(A) = 8\) and \(n(S)=52\). So \(P(A)=\frac{8}{52}=\frac{2}{13}\).

Answer:

\(\frac{2}{13}\)