Question

1. what are the odds in favour of drawing the following from a standard deck of cards?

a) a 10
b) a spade
c) a black king
d) a red face card

Explanation:

Step1: Recall deck - card details

A standard deck has 52 cards.

Step2: Calculate odds formula

The odds in favour of an event $E$ is given by $\frac{n(E)}{n(\text{not }E)}$, where $n(E)$ is the number of favorable outcomes and $n(\text{not }E)$ is the number of non - favorable outcomes.

a) For drawing a 10

Step1: Find number of 10s

There are 4 cards with the number 10 in a deck. So $n(10)=4$, and $n(\text{not }10)=52 - 4=48$.

Step2: Calculate odds

The odds in favour of drawing a 10 is $\frac{4}{48}=\frac{1}{12}$.

b) For drawing a spade

Step1: Find number of spades

There are 13 spades in a deck. So $n(\text{spade}) = 13$, and $n(\text{not spade})=52 - 13 = 39$.

Step2: Calculate odds

The odds in favour of drawing a spade is $\frac{13}{39}=\frac{1}{3}$.

c) For drawing a black king

Step1: Find number of black kings

There are 2 black kings (king of spades and king of clubs) in a deck. So $n(\text{black king}) = 2$, and $n(\text{not black king})=52 - 2=50$.

Step2: Calculate odds

The odds in favour of drawing a black king is $\frac{2}{50}=\frac{1}{25}$.

d) For drawing a red face card

Step1: Find number of red face cards

There are 6 red face cards (3 face cards in hearts and 3 face cards in diamonds: Jack, Queen, King). So $n(\text{red face card}) = 6$, and $n(\text{not red face card})=52 - 6 = 46$.

Step2: Calculate odds

The odds in favour of drawing a red face card is $\frac{6}{46}=\frac{3}{23}$.

Answer:

a) $\frac{1}{12}$
b) $\frac{1}{3}$
c) $\frac{1}{25}$
d) $\frac{3}{23}$