**The Multiplication Rule for Probability**

When we calculate the probability of one event *and *another event occurring, we multiply their probabilities.

If the outcome of the first event impacts the probability of the second event, these two events are **dependent events**.

If the outcome of the first event does not impact the probability of the second event, these two events are **independent events**.

**Multiplication Rule for ****Independent Events**

When events A and B are independent, we can find the probability that both events occur using the formula:

P(A and B) = P(A) * P(B)

**Example 1: You flip a coin twice. What is the probability that the coin lands on heads both times?**

The outcome of the first flip does not impact the probability of the second flip. No matter which side the coin lands on during the first flip, the probability that the coin lands on heads on the second flip is still 50%.

So the probability that the coin lands on heads both times is:

P(first flip is heads *and *second flip is heads) = P(first flip is heads) * P(second flip is heads)

P(first flip is heads *and *second flip is heads) = 0.5 * 0.5 = **0.25**

The probability that the coin lands on heads both times is 0.25, or 25%.

**Example 2: You toss a die, then flip a coin. What is the probability that the die lands on a four and the coin lands on tails?**

The outcome of the die toss does not affect the outcome of the coin flip, which means these two events are independent and we can simply multiply their probabilities together:

P(four and tails) = P(four) * P(tails)

P(four and tails) = (1/6) * (1/2) = **1/12**

When the outcomes of successive events are independent and we want to know the probability of some event happening *n *times in a row, we use the following formula:

P(event)^{n}

**Example 3: The probability that a coin lands on heads on two flips in a row is:**

(0.5)^{2} = **0.25**

**Example 4: The probability that a coin lands on heads on four flips in a row is:**

(0.5)^{4} = **0.0625**

**Example 5: Suppose a basketball player has a 70% free throw shooting percentage. This means the probability that he makes any given free throw is 0.70. The probability that this player makes five free throws in a row is:**

(0.70)^{5} = **0.16807**

**Multiplication Rule for Dependent Events**

When the outcome of event A impacts the probability of event B, we say that these two events are dependent. To find the probability that both events occur, we use the following formula:

P(A and B) = P(A) * P(B | A)

*Note:* P(B | A) means “the probability that B occurs, given that A has occurred.”

**Example 1: There are 3 red marbles and 7 blue marbles in a bag. You reach into the bag and randomly grab one marble. Without putting it back, you reach in and randomly grab another marble. What is the probability that you grab a red marble the first time and a blue marble the second time?**

P(red marble and blue marble) = P(red marble) * P(blue marble given that red marble was grabbed)

P(red marble and blue marble) = (3/10) * (7/9) = (21/90) = **7/30**

**Explanation:** The first time you reach in the bag, there are 10 marbles, 3 of which are blue. So the probability that you grab a blue one the first time is 3/10. The second time you reach in the bag, there are only 9 marbles, 7 of which are red. So the probability that you grab a red one the second time is 7/9. To find the probability that we grab a blue marble first, then a red one, we simply multiply these two probabilities together.

**Example 2:** **A classroom has 12 sophomores and 8 freshman. A teacher selects two students at random to present homework problems. What is the probability that both students selected are freshman?**

P(both freshman) = P(freshman selected first) * (freshman selected second)

P(both freshman) = (8/20) * (7/19) = (56/380) = 14/95

**Explanation:** The probability that the teacher randomly selects a freshman the first time is 8/20. Then, once that freshman has been selected, there are 7 freshman left out of the 19 remaining students. So the probability that the teacher randomly selects a freshman the second time is 7/19. Then we simply multiply these two probabilities together.

**Multiplication Rule for “At Least One” Success**

To find the probability of at least one success in a series of independent events, we use the formula:

P(at least 1 success) = 1 − P(all failures)

Similarly, to find the probability of at least one failure in a series of independent events, we use the formula:

P(at least one failure) = 1 – P(all successes)

**Example 1: Dirk makes 85% of the free throws that he attempts. The results of each attempt are independent. If he makes four attempts, what is the probability that he misses at least one?**

P(misses at least one attempt) = 1 – P(makes every attempt)

P(misses at least one attempt) = 1 – (0.85)^{4} =** 0.478**

**Example 2: Suppose 20% of students in a particular class prefer pizza over ice cream. If we select three students at random, what is the probability that at least one student prefers pizza over ice cream?**

P(at least one prefers pizza) = 1 – P(none prefer pizza)

P(at least one prefers pizza) = 1 – (0.8)^{3} =** 0.488**