two six - sided dice are tossed. event a: the first die lands on 1, 2, 3, or 4. event b: the second die…

two six - sided dice are tossed. event a: the first die lands on 1, 2, 3, or 4. event b: the second die lands on 3, 4, 5, or 6. what is the probability that both events will occur? for independent events: p(a and b)=p(a)·p(b) p(a and b)=? give your answer in simplest form.

two six - sided dice are tossed. event a: the first die lands on 1, 2, 3, or 4. event b: the second die lands on 3, 4, 5, or 6. what is the probability that both events will occur? for independent events: p(a and b)=p(a)·p(b) p(a and b)=? give your answer in simplest form.

Answer

Explanation:

Step1: Calculate P(A)

The first die has 6 possible outcomes. Event A has 4 favorable outcomes (1, 2, 3, 4). So $P(A)=\frac{4}{6}=\frac{2}{3}$.

Step2: Calculate P(B)

The second die has 6 possible outcomes. Event B has 4 favorable outcomes (3, 4, 5, 6). So $P(B)=\frac{4}{6}=\frac{2}{3}$.

Step3: Calculate P(A and B)

Since A and B are independent events, $P(A\ and\ B)=P(A)\cdot P(B)$. Substitute $P(A)=\frac{2}{3}$ and $P(B)=\frac{2}{3}$ into the formula. So $P(A\ and\ B)=\frac{2}{3}\times\frac{2}{3}=\frac{4}{9}$.

Answer:

$\frac{4}{9}$