A box contains a one-dollar bill, a five-dollar bill, a ten-dollar bill, and a twenty-dollar bill. Two bills are chosen in succession without replacement. Use a tree diagram to list the sample space for this experiment and then answer the following questions in fractions
1.
What is the probability that the values of both bills are even?
2.
What is the probability that the value of neither bill is even?
3.
What is the probability that the value of exactly one of the bills is even?
4.
What is the probability that the value of at least one of the bills is even?
5.
What is the probability that the total value of the bills chosen is $30?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
1. $10 and $20 are even. The chances at each drawing are 1/2.
2. $1 and $5 are odd. The chances at each drawing are 1/2.
3. Same.
4. P(1 or 2 bills even). Either-or probabilities are found by adding the individual probabilities.
5. see #1
no
To construct a tree diagram for this experiment, we can list all possible outcomes for the first bill, and then for each of those outcomes, list all possible outcomes for the second bill.
The sample space for this experiment can be represented by the following simplified tree diagram:
1 ($)---5 ($)---10 ($)---20 ($)
| (odd) | (odd) | (odd)
| | |
| (odd) | (odd) |
| | |
1 ($)---5 ($)---10 ($)---20 ($)
| (odd) | (odd) | (even) |
| | | |
| (odd) | (even) | |
| | | |
| (even) | | |
1. The probability that both bills have even values can be calculated by counting the number of favorable outcomes (even values for both bills) and dividing it by the total number of possible outcomes.
Favorable outcomes: There is only one favorable outcome - starting with the 10-dollar bill and then choosing the 20-dollar bill.
Total outcomes: There are a total of four possible outcomes, as each bill can be chosen first.
Probability = Favorable outcomes / Total outcomes = 1/4
Therefore, the probability that the values of both bills are even is 1/4.
2. The probability that neither bill has an even value can be calculated in the same way, by counting the number of favorable outcomes and dividing it by the total number of possible outcomes.
Favorable outcomes: There are two favorable outcomes - starting with the 1-dollar bill and then choosing the 5-dollar bill or starting with the 5-dollar bill and then choosing the 1-dollar bill.
Total outcomes: There are a total of four possible outcomes, as each bill can be chosen first.
Probability = Favorable outcomes / Total outcomes = 2/4 = 1/2
Therefore, the probability that neither bill has an even value is 1/2.
3. The probability that exactly one of the bills has an even value can be calculated in the same way, by counting the number of favorable outcomes and dividing it by the total number of possible outcomes.
Favorable outcomes: There are two favorable outcomes - starting with the 1-dollar bill and then choosing the 10-dollar bill, or starting with the 10-dollar bill and then choosing the 1-dollar bill.
Total outcomes: There are a total of four possible outcomes, as each bill can be chosen first.
Probability = Favorable outcomes / Total outcomes = 2/4 = 1/2
Therefore, the probability that exactly one of the bills has an even value is 1/2.
4. The probability that at least one of the bills has an even value can be calculated by counting the number of favorable outcomes (any outcome except both bills being odd) and dividing it by the total number of possible outcomes.
Favorable outcomes: There are three favorable outcomes - starting with the 1-dollar bill and then choosing the 5-dollar bill, starting with the 5-dollar bill and then choosing the 1-dollar bill, or starting with the 10-dollar bill and then choosing the 20-dollar bill.
Total outcomes: There are a total of four possible outcomes, as each bill can be chosen first.
Probability = Favorable outcomes / Total outcomes = 3/4
Therefore, the probability that at least one of the bills has an even value is 3/4.
5. The probability that the total value of the bills chosen is $30 can be calculated by counting the number of favorable outcomes (choosing the 10-dollar bill and then the 20-dollar bill) and dividing it by the total number of possible outcomes.
Favorable outcomes: There is only one favorable outcome - starting with the 10-dollar bill and then choosing the 20-dollar bill.
Total outcomes: There are a total of four possible outcomes, as each bill can be chosen first.
Probability = Favorable outcomes / Total outcomes = 1/4
Therefore, the probability that the total value of the bills chosen is $30 is 1/4.
To answer these questions, first, let's draw a tree diagram to list the sample space for this experiment:
START
/ \
1 5 10 20
/|\ /|\ /|\ /|\
5 10 20 1 10 20 1 5 20 1 5 10
Now, let's answer the questions:
1. What is the probability that the values of both bills are even?
To find the probability that both bills have even values, we need to find the number of desired outcomes and divide it by the total number of possible outcomes. In this case, there are three even bills in the box: 2, 10, and 20. So the number of desired outcomes is 3. The total number of possible outcomes is 4, as there are four bills in total. Therefore, the probability is 3/4.
2. What is the probability that the value of neither bill is even?
To find the probability that neither bill has an even value, we need to find the number of desired outcomes (no even bills) and divide it by the total number of possible outcomes. In this case, there is only one bill with an odd value, which is the 1-dollar bill. So the number of desired outcomes is 1. Since there are four bills in total, the probability is 1/4.
3. What is the probability that the value of exactly one of the bills is even?
To find the probability that exactly one bill has an even value, we need to find the number of desired outcomes (one even bill and one odd bill) and divide it by the total number of possible outcomes. There are three possible scenarios: (1, 5), (1, 10), and (1, 20). So the number of desired outcomes is 3. The total number of possible outcomes is still 4. Therefore, the probability is 3/4.
4. What is the probability that the value of at least one of the bills is even?
To find the probability that at least one bill has an even value, we need to find the number of desired outcomes (cases where at least one bill has an even value) and divide it by the total number of possible outcomes. In this case, there are two desired outcomes: (1, 5) and (1, 10). So the number of desired outcomes is 2. The total number of possible outcomes is 4. Therefore, the probability is 2/4, which simplifies to 1/2.
5. What is the probability that the total value of the bills chosen is $30?
To find the probability that the total value of the bills chosen is $30, we need to find the number of desired outcomes (cases where the sum of the values is $30) and divide it by the total number of possible outcomes. In this case, there is only one desired outcome: (10, 20). So the number of desired outcomes is 1. The total number of possible outcomes is 4. Therefore, the probability is 1/4.
Remember to always use the formula: probability = desired outcomes / total outcomes to find the probabilities.