Question

question 9
4.2 theoretical probability. calculating theoretical probabilities (deck of cards).
an experiment is to draw 1 card from a fair deck of playing cards.
a) if you were to list the sample space, how many equally likely outcomes would it have?
b) find the probability of getting a 6.
leave your answer as a fraction.
c) find the probability of not getting a diamond.
leave your answer as a fraction.
d) find the probability of getting a spade.
leave your answer as a fraction.
e) find the probability of getting a 6 or a heart.
leave your answer as a fraction.
question help: video 1 video 2 message instructor

Explanation:

Step1: Determine sample - space size

A standard deck of playing cards has 52 cards. So the number of equally - likely outcomes in the sample space is 52.

Step2: Calculate probability of getting a 6

There are 4 cards with the number 6 in a deck (one 6 in each suit). Using the probability formula $P(A)=\frac{n(A)}{n(S)}$, where $n(A)$ is the number of favorable outcomes and $n(S)$ is the number of outcomes in the sample space. Here, $n(A) = 4$ and $n(S)=52$, so $P(\text{6})=\frac{4}{52}=\frac{1}{13}$.

Step3: Calculate probability of not getting a diamond

There are 13 diamonds in a deck. So the number of non - diamond cards is $52 - 13=39$. Then $P(\text{not diamond})=\frac{39}{52}=\frac{3}{4}$.

Step4: Calculate probability of getting a spade

There are 13 spades in a deck. Using the probability formula, $P(\text{spade})=\frac{13}{52}=\frac{1}{4}$.

Step5: Calculate probability of getting a 6 or a heart

Let $A$ be the event of getting a 6 and $B$ be the event of getting a heart. $n(A) = 4$, $n(B)=13$, and $n(A\cap B) = 1$ (the 6 of hearts). Using the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$, we have $P(A)=\frac{4}{52}$, $P(B)=\frac{13}{52}$, and $P(A\cap B)=\frac{1}{52}$. Then $P(A\cup B)=\frac{4 + 13-1}{52}=\frac{16}{52}=\frac{4}{13}$.

Answer:

a) 52
b) $\frac{1}{13}$
c) $\frac{3}{4}$
d) $\frac{1}{4}$
e) $\frac{4}{13}$