Question

A card is drawn from a pack of 52 cards. A gambler bets that it is a spade or ace. What are the odds against his winning the bet?
A. 9:4
B. 4:9
C. 5:9
D. 9:5

Answer 1

Hint: First we will calculate the total number of spades and aces in a deck. Then apply the formula of probability to calculate the probability of winning. Now calculate the probability of losing the bet by using the formula \[P\left( {\overline E } \right) = 1 - P\left( E \right)\]. To calculate the required probability, find the ratio of the probability of losing to the probability of winning.

Formula Used:
\[{\rm{Probability = }}\dfrac{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{Favorable}}\,{\rm{outcomes}}}}{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{outcomes}}}}\]
\[P\left( {\overline E } \right) = 1 - P\left( E \right)\]

Complete step by step solution:
The total number of cards in a deck is 52.
The total number of aces in a deck is 4.
The total number of spades in a deck is 13.
In 13 spades, there is one ace.
So total number of spades and ace is \[\left( {13 + 4 - 1} \right) = 16\].
The number of favorable outcomes is 16.
Using the formula \[{\rm{Probability = }}\dfrac{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{Favorable}}\,{\rm{outcomes}}}}{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{outcomes}}}}\] to calculate the probability of getting a spade or ace.
\[{\rm{Probability = }}\dfrac{{{\rm{16}}}}{{{\rm{52}}}}\]
         \[ = \dfrac{4}{{13}}\]
So, the probability of winning is \[\dfrac{4}{{13}}\].
Using the formula \[P\left( {\overline E } \right) = 1 - P\left( E \right)\], we will calculate the probability of losing.
\[P\left( {\overline E } \right) = 1 - \dfrac{4}{{13}}\]
          \[ = \dfrac{{13 - 4}}{{13}}\]
          \[ = \dfrac{9}{{13}}\]
Finding the ratio of the probability of losing and winning
\[\dfrac{9}{{13}}:\dfrac{4}{{13}}\]
\[ = \dfrac{9}{{13}} \times \dfrac{{13}}{4}\]
Cancel out \[13\]
\[ = 9:4\]
Therefore, the odds against him winning the bet are 9:4.

Option ‘A’ is correct

Note: The question is asked to find the odds against winning the bet and not the probability of losing the bet, both of these probabilities are far different. For finding the probability of losing the bet do not go for a long method like P (neither, nor).
Just use the formula \[P\left( {\overline E } \right) = 1 - P\left( E \right)\] to solve