Question

independent events
to find the probability of two independent events occurring together, we multiply the probabilities of each event.
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 and an even number?
the probability of getting the heart card is $\frac{1}{4}$ and the probability of rolling an even number is $\frac{1}{2}$. to find the probability of getting the heart card and rolling an even number, we multiply these two probabilities.
so, the probability of getting a heart card and rolling an even number is $\frac{1}{2}×\frac{1}{4}=\frac{1}{8}$
try these:
a regular die has 6 sides and a coin has 2 sides. you flip the coin once and roll the die once.
what is the probability of getting:

1. heads and a 5?
2. tails and an even number?
3. heads and a 2 or a 6?

Explanation:

Step1: Calculate probability of coin - event

The probability of getting heads when flipping a coin is $\frac{1}{2}$, and the probability of getting tails is also $\frac{1}{2}$.

Step2: Calculate probability of die - event for 1)

The probability of rolling a 5 on a 6 - sided die is $\frac{1}{6}$. Since the coin - flip and die - roll are independent events, we multiply the probabilities. So for getting heads and a 5, the probability is $\frac{1}{2}\times\frac{1}{6}=\frac{1}{12}$.

Step3: Calculate probability of die - event for 2)

The probability of getting an even number (2, 4, 6) on a 6 - sided die is $\frac{3}{6}=\frac{1}{2}$. The probability of getting tails on a coin - flip is $\frac{1}{2}$. Since they are independent, the probability of getting tails and an even number is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$.

Step4: Calculate probability of die - event for 3)

The probability of rolling a 2 or a 6 on a 6 - sided die is $\frac{2}{6}=\frac{1}{3}$. The probability of getting heads on a coin - flip is $\frac{1}{2}$. Since they are independent, the probability of getting heads and a 2 or a 6 is $\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}$.

Answer:

1. $\frac{1}{12}$
2. $\frac{1}{4}$
3. $\frac{1}{6}$