Assume that there is an urn containing seven $50 bills, six $20 bills, five $10 bills, four $5 bills, and one $1 bill and that the bills all have different serial numbers so that they can be distinguished from each other. A person reaches into the urn and withdraws one bill and then another;
a) in how many ways can two $20 bills be withdrawn
b) how many different outcomes are possible
c) what is the probability of selecting two $20 bills?
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a) To solve this, we need to calculate the number of ways two $20 bills can be withdrawn.
Step 1: Selecting the first $20 bill: There are 6 $20 bills in the urn.
Step 2: Selecting the second $20 bill: There are also 6 $20 bills remaining in the urn since we assume that each bill has a unique serial number and can be distinguished from each other.
So the total number of ways two $20 bills can be withdrawn is 6 x 6 = 36 ways.
b) To calculate the different outcomes, we need to find the total number of ways any two bills can be withdrawn from the urn.
The total number of bills in the urn is:
7 $50 bills + 6 $20 bills + 5 $10 bills + 4 $5 bills + 1 $1 bill = 23 bills
To calculate the different outcomes, we need to find all possible combinations of choosing 2 bills from the 23 bills.
The formula for calculating combinations is:
C(n, k) = n! / (k!(n - k)!)
Where n is the number of items to choose from, and k is the number of items to choose.
In our case, n = 23 (since there are 23 bills in the urn), and k = 2.
So the number of different outcomes is:
C(23, 2) = 23! / (2!(23 - 2)!) = 23! / (2!21!) = (23 x 22) / (2 x 1) = 253
Therefore, there are 253 different outcomes possible.
c) To calculate the probability of selecting two $20 bills, we need to find the ratio of the number of successful outcomes (in this case, selecting two $20 bills) to the total number of possible outcomes.
The number of successful outcomes is 36 (as calculated in part a).
The total number of possible outcomes was found to be 253 (as calculated in part b).
So the probability of selecting two $20 bills is:
Probability = Number of Successful Outcomes / Total Number of Outcomes
= 36 / 253
= 0.1421 or approximately 14.21%
To find the answers to these questions, we can use the concept of combinations and the counting principle of probability.
a) In how many ways can two $20 bills be withdrawn?
To calculate the number of ways two $20 bills can be withdrawn, we need to find the combination of selecting two $20 bills from the given set. The number of ways to select two $20 bills from a set of six is given by the formula:
C(n, r) = n! / [(n-r)! * r!]
where n is the total number of elements in the set, and r is the number of elements we want to select.
In this case, there are six $20 bills in the urn. So the number of ways to select two $20 bills is:
C(6, 2) = 6! / [(6-2)! * 2!] = 6! / 4! * 2! = (6 * 5 * 4!) / (4! * 2 * 1) = 15
Therefore, there are 15 ways to withdraw two $20 bills from the urn.
b) How many different outcomes are possible?
To find the total number of different outcomes, we need to consider all possible combinations of bills that can be withdrawn. This includes combinations with different denominations.
The total number of different outcomes is calculated by summing up the number of ways to select 2, 1, or 0 bills of each denomination:
- Number of ways to select 2 bills of the same denomination = C(n, 2)
- Number of ways to select exactly 1 bill of a certain denomination = n (number of bills of that denomination)
- Number of ways to select 0 bills of a certain denomination = 1 (as it leaves all of them in the urn)
Let's calculate the total number of different outcomes:
Number of ways to select 2 $50 bills = C(7, 2) = 7! / (5! * 2!) = 21
Number of ways to select 2 $20 bills = C(6, 2) = 15
Number of ways to select 2 $10 bills = C(5, 2) = 10
Number of ways to select 2 $5 bills = C(4, 2) = 6
Number of ways to select 2 $1 bills = C(1, 2) = 0 (not possible)
Total number of different outcomes = 21 + 15 + 10 + 6 + 0 = 52
Therefore, there are 52 different outcomes possible.
c) What is the probability of selecting two $20 bills?
To find the probability, we need to divide the number of favorable outcomes (selecting two $20 bills) by the total number of outcomes.
Number of favorable outcomes = 15 (as calculated in part a)
Total number of outcomes = 52 (as calculated in part b)
Probability of selecting two $20 bills = Number of favorable outcomes / Total number of outcomes
= 15 / 52
= 0.288 or approximately 28.8%
Therefore, the probability of selecting two $20 bills is approximately 0.288 or 28.8%.