Question

• probability and tree diagrams •

lets look at the previous example with a slight difference. this time we are looking for the probability of getting the heart card or an even number.
example: if 1 of the 4 cards is picked at random and the die is rolled once, what is the probability of getting the heart card or an even number?
first we can make a tree.
we know that there are 24 possible outcomes. of those, 15 are the \winning\ combinations that contain either a heart or an even number: (heart, 1), (heart, 2), (heart, 3), (heart, 4), (heart, 5), (heart, 6), (club, 2), (club, 4), (club, 6), (diamond, 2), (diamond, 4), (diamond, 6), (spade, 2), (spade, 4), and (spade, 6), circled above. so the probability of getting the heart card or an even number is $\frac{15}{24}$ or $\frac{5}{8}$.
try these: use the probability tree in the example above.
what is the probability of:

1. picking the spade card or rolling an even number? $\frac{15}{24}$
2. picking a \black card\ (club or spade) or rolling more than 3?
3. not picking the club card or not rolling a 6? $\frac{23}{24}$
4. not picking the spade card or not rolling a factor of 6?

Explanation:

Step1: Calculate total outcomes

There are 4 cards and 6 - sided die, so total outcomes = 4×6 = 24.

Step2: Calculate favorable outcomes for picking spade or rolling even

Number of spade - related outcomes: (spade, 1), (spade, 2), (spade, 3), (spade, 4), (spade, 5), (spade, 6) = 6.
Number of non - spade and even outcomes: (heart, 2), (heart, 4), (heart, 6), (club, 2), (club, 4), (club, 6), (diamond, 2), (diamond, 4), (diamond, 6)=9.
Total favorable outcomes = 6 + 9=15.
Probability = $\frac{15}{24}=\frac{5}{8}$.

Step3: Calculate favorable outcomes for picking black card or rolling more than 3

Number of black - card related outcomes: (club, 1), (club, 2), (club, 3), (club, 4), (club, 5), (club, 6), (spade, 1), (spade, 2), (spade, 3), (spade, 4), (spade, 5), (spade, 6)=12.
Number of non - black and rolling more than 3 outcomes: (heart, 4), (heart, 5), (heart, 6), (diamond, 4), (diamond, 5), (diamond, 6)=6.
Total favorable outcomes = 12+6 = 18.
Probability = $\frac{18}{24}=\frac{3}{4}$.

Step4: Calculate favorable outcomes for not picking club or not rolling 6

Number of not - club outcomes: (heart, 1), (heart, 2), (heart, 3), (heart, 4), (heart, 5), (heart, 6), (diamond, 1), (diamond, 2), (diamond, 3), (diamond, 4), (diamond, 5), (diamond, 6), (spade, 1), (spade, 2), (spade, 3), (spade, 4), (spade, 5), (spade, 6)=18.
Number of club and not 6 outcomes: (club, 1), (club, 2), (club, 3), (club, 4), (club, 5)=5.
Total favorable outcomes = 18 + 5=23.
Probability = $\frac{23}{24}$.

Step5: Calculate favorable outcomes for not picking spade or not rolling factor of 6

Factors of 6 are 1, 2, 3, 6.
Number of not - spade outcomes: (heart, 1), (heart, 2), (heart, 3), (heart, 4), (heart, 5), (heart, 6), (club, 1), (club, 2), (club, 3), (club, 4), (club, 5), (club, 6), (diamond, 1), (diamond, 2), (diamond, 3), (diamond, 4), (diamond, 5), (diamond, 6)=18.
Number of spade and non - factor of 6 outcomes: (spade, 4), (spade, 5)=2.
Total favorable outcomes = 18+2 = 20.
Probability = $\frac{20}{24}=\frac{5}{6}$.

Answer:

1. $\frac{5}{8}$
2. $\frac{3}{4}$
3. $\frac{23}{24}$
4. $\frac{5}{6}$