The signers of the declaration of Independence came from the 13 colonies as shown:
Massachusetts: 5
New Hampshire: 3
Virginia: 7
Maryland: 4
New Jersey: 5
Pennsylvania: 9
Rhode Island: 2
New York: 4
Georgia: 3
North Carolina: 3
South Carolina: 4
Connecticut: 4
Delaware: 3
Suppose that 4 are chosen at random. Find the probability that 2 come from Pennsylvania and 2 come from Virginia
I'm not sure where to start, but I do know we've been using Permutations and Combinations.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
9/56 * (9-1)/(56-1) * 7/(56-2) * (7-1)/56-3) = ?
Which formula would fit best?
To solve this problem, we need to use combinations, which are a way to count the number of possible outcomes when order does not matter. In this case, we want to find the probability of selecting 2 signers from Pennsylvania and 2 signers from Virginia, regardless of the order in which they are selected.
First, let's count the total number of ways to choose any 4 signers from the 13 colonies. We can use combinations to do this. The formula for combinations is given by C(n, r) = n! / (r!(n-r)!), where n is the total number of items to choose from, and r is the number of items to select.
In our case, we are choosing 4 signers from 13 colonies, so n = 13 and r = 4. Using the combination formula, we have:
C(13, 4) = 13! / (4!(13-4)!)
= 13! / (4! * 9!)
The exclamation point denotes factorial, which means the product of all positive integers less than or equal to the number. For example, 4! = 4 * 3 * 2 * 1.
Now, let's count the number of ways to choose 2 signers from Pennsylvania and 2 signers from Virginia. Since there are 9 signers from Pennsylvania and 7 signers from Virginia, we have:
C(9, 2) * C(7, 2) = (9! / (2! * 7!)) * (7! / (2! * 5!))
Simplifying this expression, we get:
C(9, 2) * C(7, 2) = (9 * 8 / 2 * 1) * (7 * 6 / 2 * 1)
= (36) * (21)
= 756
So there are 756 ways to choose 2 signers from Pennsylvania and 2 signers from Virginia.
Now, to find the probability, we need to divide the number of favorable outcomes (756) by the number of total outcomes (C(13, 4)). Therefore, the probability is:
P(2 from Pennsylvania and 2 from Virginia) = 756 / C(13, 4)