Question

Set up an appropriate sample space for each of Problems 1.1 to 1.10 and use itto solve the problem. Use either a uniform or non-uniform sample space or try both.

A trick deck of cards is printed with the hearts and diamonds black, and the spadesand clubs red. A card is chosen at random from this deck (after it is shuffled). Findthe probability that it is either a red card or the queen of hearts. That it is eithera red face card or a club. That it is either a red ace or a diamond.

Short answer

The required sample space for the given problem is all the cards.

The probability of a card drawn is either a red card or the queen of hearts is $\frac{27}{52}$.

The probability that the card drawn is either a red face card or a club is $\frac{4}{13}$.

The probability that the card drawn is either a red ace or a diamond is$\frac{15}{52}$.

Step-by-step solution

Definition of trick deck of card 

The deck of cards in which the cards are adjusted in a different way for playing some tricks with cards. In number of cards remain same, that is 52. The cards are mostly tricked for performing magic tricks.

Creation of Sample Space

In a deck of card, there are 52 cards of 4 suits namely spade, club, diamond and heart, out of which spade and club are black and diamond and heart are red.

There are 13 cards of each suit and 26 cards of each color.

In the tricked deck, there are 52 cards of 4 suits namely spade, club, diamond and heart, out of which spade and club are red and diamond and heart are black.

There are 13 cards of each suit and 26 cards of each color.

The Sample space is all the cards, that is 52.

Determination of the probability that is either a red card or the queen of hearts

Each point of the obtained sample space has an equal probability of $\frac{1}{52}$

In this deck of card, there is 1 queen of hearts(which is no red) and 26 red cards, this implies that total number of possibility for a card to be an queen of heart or red is expressed as follows,

$1+26=27$

Find the probability that the card drawn is either a red card or the queen of hearts by adding the probabilities of each possible outcomes, that is 27 times $\frac{\mathbf{1}}{\mathbf{52}}$.

$p=27\times \frac{1}{52}$

Thus, the probability of a card drawn is either a red card or the queen of hearts is$\frac{27}{52}$.

Determination ofthe probability that it is either a red face card or a club

In a deck of card, there are 13 club cards(which are red) and 6 red face cards out of which 3are of clubs, this implies that total number of possibility for a card to be a red face card or a club is expressed as follows,

$13+6-3=16$

Find the probability that the card drawn is either a red face card or a clubby adding the probabilities of each possible outcomes, that is 16 times$\frac{\mathbf{1}}{\mathbf{52}}$.

$\begin{array}{rcl}p& =& 16\times \frac{1}{52}\\ & =& \frac{4}{13}\\ & & \end{array}$

Thus, the probability that the card drawn is either a red face card or a club is$\frac{4}{13}$.

Determination ofthe probability that the card drawn is either a red ace or a diamond

In this deck of card, there are 2 red aces and 13 diamond cards, this implies that total number of possibility for a card to be either a red ace or a diamondis$2+13=15$

Find the probability that the card drawn is either a red ace or a diamond by adding the probabilities of each possible outcomes, that is 15 times $\frac{\mathbf{1}}{\mathbf{52}}$

$p=15\times \frac{1}{52}$

Thus, the probability that the card drawn is either a red ace or a diamond is$\frac{15}{52}$