Listed below is the population by state for the 15 states with the largest population. Also included is whether that state's border touches the Gulf of Mexico, the Atlantic Ocean, or the Pacific Ocean (coastline).
Rank State Population Coastline
1 California 36,553,215 Yes
2 Texas 23,904,380 Yes
3 New York 19,297,729 Yes
4 Florida 18,251,243 Yes
5 Illinois 12,852,548 No
6 Pennsylvania 12,432,792 No
7 Ohio 11,466,917 No
8 Michigan 10,071,822 No
9 Georgia 9,544,750 Yes
10 North Carolina 9,061,032 Yes
11 New Jersey 8,685,920 Yes
12 Virginia 7,712,091 Yes
13 Washington 6,468,424 Yes
14 Massachusetts 6,449,755 Yes
15 Indiana 6,345,289 No
Note that 5 of the 15 states do not have any coastline. Suppose three states are selected at random. Calculate the probabilities for the following.
(a) None of the states selected have any coastline (Round your answer to 3 decimal places.)
(b) Exactly one of the selected states has a coastline (Round your answer to 2 decimal places.)
(c) At least one of the selected states has a coastline (Round your answer to 3 decimal places.)
;gtb
To calculate the probabilities for the given scenarios, we need to determine the total number of possible outcomes and the favorable outcomes for each scenario.
First, let's calculate the total number of possible outcomes when selecting three states out of the fifteen. We can use the combination formula, also known as "n choose r", to calculate this. The formula is:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items to be chosen. In this case, n = 15 (the total number of states) and r = 3 (the number of states to be selected).
Using the formula, we have:
C(15, 3) = 15! / (3!(15-3)!)
= 15! / (3!12!)
= (15 * 14 * 13 * 12!) / (3!12!)
= (15 * 14 * 13) / (3 * 2 * 1)
= 455
So, there are a total of 455 possible outcomes when selecting three states out of the fifteen.
(a) To calculate the probability that none of the states selected have any coastline, we need to determine the favorable outcomes, which are the number of combinations that include only the 5 states without coastline (Illinois, Pennsylvania, Ohio, Michigan, Indiana).
The number of combinations selecting only the states without coastline is given by:
C(5, 3) = 5! / (3!(5-3)!)
= 5! / (3!2!)
= (5 * 4 * 3!) / (3!2!)
= 10
Therefore, there are 10 combinations that include only states without coastline. The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
P(None of the states selected have any coastline) = 10 / 455 ≈ 0.022
So, the probability that none of the states selected have any coastline is approximately 0.022.
(b) To calculate the probability that exactly one of the selected states has a coastline, we need to determine the favorable outcomes, which are the number of combinations that include one state with coastline and two states without coastline.
The number of combinations selecting one state with coastline is given by:
C(5, 1) = 5! / (1!(5-1)!)
= 5! / (1!4!)
= 5
The number of combinations selecting two states without coastline is given by:
C(10, 2) = 10! / (2!(10-2)!)
= 10! / (2!8!)
= (10 * 9) / (2 * 1)
= 45
Therefore, there are 5 * 45 = 225 combinations that include exactly one state with coastline and two states without coastline. The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
P(Exactly one of the selected states has a coastline) = 225 / 455 ≈ 0.495
So, the probability that exactly one of the selected states has a coastline is approximately 0.495.
(c) To calculate the probability that at least one of the selected states has a coastline, we can use the complement rule. The complement of "at least one state with coastline" is "no state with coastline".
P(At least one of the selected states has a coastline) = 1 - P(None of the states selected have any coastline)
From part (a), we calculated that the probability of selecting no state with coastline is approximately 0.022. Therefore:
P(At least one of the selected states has a coastline) = 1 - 0.022 = 0.978
So, the probability that at least one of the selected states has a coastline is approximately 0.978.
In summary:
(a) The probability that none of the states selected have any coastline is approximately 0.022.
(b) The probability that exactly one of the selected states has a coastline is approximately 0.495.
(c) The probability that at least one of the selected states has a coastline is approximately 0.978.