A nickel is flipped and a dime is flipped.

1. What is the probability of obtaining exactly one head?

A bag contains five balls numbered 0 through 4. Two balls are chosen at random in succession. The first ball is replaced before the second is chosen.

1. What is the probability that the first ball has an odd number?

2. What is the probability that the second ball has an odd number?

3. What is the probability that both balls have odd numbers?

4. What is the probability that neither ball has an odd number?

1. The possible outcomes are HH, TT, TH and HT. What do you think?

Balls

1. In 0, 1, 2, 3 and 4, only 1 and 3 are odd.

2. Same info.

3. If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

4. 0, 2 and 4 are not odd. Use rule from #3.

A nickel and a quarter are flipped in succession. Given that the nickel landed on tails, what is the probability that the quarter lands on heads?

1. To find the probability of obtaining exactly one head when flipping a nickel and a dime, we can use the concept of probability multiplication. Since each flip is independent, the probability of getting a head on the nickel is 1/2, and the probability of getting a head on the dime is also 1/2.

To obtain exactly one head, we can have either the nickel showing heads and the dime showing tails, or the nickel showing tails and the dime showing heads. Since these two outcomes are mutually exclusive, we can add their probabilities. Therefore, the probability of obtaining exactly one head is:

P(1 head) = P(nickel heads) * P(dime tails) + P(nickel tails) * P(dime heads)
= (1/2) * (1/2) + (1/2) * (1/2)
= 1/4 + 1/4
= 1/2
= 0.5 (or 50%)

2. To find the probability that the first ball has an odd number when choosing two balls with replacement from a bag of five balls numbered 0 through 4, we need to determine the number of favorable outcomes (odd numbers) and divide it by the total number of possible outcomes (all numbers 0 through 4).

Since there are three odd numbers (1, 3, and 4) out of the five total numbers, the probability of choosing a ball with an odd number is:

P(odd number on the first ball) = Number of odd numbers / Total number of possible outcomes
= 3 / 5
= 0.6 (or 60%)

3. To find the probability that the second ball has an odd number, we use the same logic as in the previous question, considering that we are replacing the first ball before choosing the second one.

Since there are three odd numbers (1, 3, and 4) out of the five total numbers, the probability of choosing a ball with an odd number is:

P(odd number on the second ball) = Number of odd numbers / Total number of possible outcomes
= 3 / 5
= 0.6 (or 60%)

4. To find the probability that both balls have odd numbers, we need to multiply the probabilities of obtaining an odd number on the first ball and an odd number on the second ball because these events are independent.

P(both balls have odd numbers) = P(odd number on the first ball) * P(odd number on the second ball)
= 0.6 * 0.6
= 0.36 (or 36%)

5. To find the probability that neither ball has an odd number, we can subtract the probability of both balls having odd numbers from 1, because if both balls have an odd number, then neither ball can have an even number.

P(neither ball has an odd number) = 1 - P(both balls have odd numbers)
= 1 - 0.36
= 0.64 (or 64%)

To find the probability of an event, we need to know the total number of possible outcomes and the number of favorable outcomes.

In the case of flipping a nickel and a dime, there are four possible outcomes for each coin (Heads or Tails), resulting in a total of 4 x 4 = 16 possible outcomes for both coins combined.

1. The probability of obtaining exactly one head can be calculated by counting the favorable outcomes. Here, there are two favorable outcomes: nickel (Heads) and dime (Tails), and dime (Heads) and nickel (Tails). Therefore, the probability is 2/16 = 1/8.

Now let's move on to the bag with numbered balls.

1. To calculate the probability that the first ball has an odd number, we need to know the total number of balls (5) and the number of odd-numbered balls (3). So the probability is 3/5.

2. As the first ball is replaced, the probability that the second ball has an odd number remains the same. So again, the probability is 3/5.

3. To calculate the probability that both balls have odd numbers, we multiply the probabilities of the individual events. So the probability of the first ball being odd is 3/5, and the probability of the second ball being odd, considering the replacement, is also 3/5. Therefore, the probability is (3/5) x (3/5) = 9/25.

4. To calculate the probability that neither ball has an odd number, we need to find the probability of both balls being even. There are two even-numbered balls (0 and 4). So the probability of the first ball being even is 2/5, and the probability of the second ball being even, considering the replacement, is also 2/5. Therefore, the probability is (2/5) x (2/5) = 4/25.