At a summer camp, 20% of the students come from out of state. Megan wants to simulate cabins of 8 campers. She generates random digits from 0 to 9 and lets the digits 0 and 1 represent a camper from out of state. The table below shows her results.
69531217
89542756
89001254
01346895
12468503
20312346
79564328
59868542
75891003
56103249
58630126
57498510
76134860
52974168
03164985
03289642
68533236
94210145
70215789
32605254
What is the experimental probability that in a cabin of 8 campers, none is from out of state?
A. Fraction is 1 over 20
B. one-tenth
C. one-fifth
D. one-fourth..
To determine the experimental probability that in a cabin of 8 campers, none is from out of state, we need to count the number of times Megan's random digits did not include 0 or 1. Looking through the table, we see that the following entries meet this criteria:
69531217
12468503
20312346
52974168
That is a total of 4 out of 20 entries. So the experimental probability is:
4/20 = 1/5
Therefore, the answer is C. One-fifth.
To find the experimental probability that in a cabin of 8 campers, none is from out of state, we need to determine the number of combinations where none of the campers are from out of state, and then divide it by the total number of combinations.
From the given results, we can see that the digits 0 and 1 represent campers from out of state. Therefore, we need to count the number of combinations that do not have these digits.
Looking at the provided numbers, we can see that there are only two numbers that contain the digits 0 and 1: 89001254 and 94210145. Therefore, there are 18 numbers that do not contain the digits 0 and 1.
Next, we need to calculate the number of combinations of 8 campers that can be formed from these 18 numbers. To do this, we can use the combination formula, which is given by:
nCr = n! / [(n-r)! * r!]
In this case, n = 18 (the number of numbers that do not contain the digits 0 and 1) and r = 8 (the number of campers in a cabin).
Using the combination formula, we can calculate this value:
18C8 = 18! / [(18-8)! * 8!]
= 18! / (10! * 8!)
= (18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
= 43758
Therefore, there are 43,758 combinations where none of the campers in a cabin of 8 are from out of state.
The total number of combinations of 8 campers is given by 20C8, which we can calculate in the same way:
20C8 = 20! / [(20-8)! * 8!]
= 20! / (12! * 8!)
= (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13) / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
= 125970
Therefore, there are 125,970 total combinations of 8 campers.
To find the experimental probability, we divide the number of combinations where none of the campers are from out of state (43,758) by the total number of combinations (125,970).
Experimental probability = 43758 / 125970
≈ 0.347
Therefore, the experimental probability that in a cabin of 8 campers, none is from out of state is approximately 0.347, which can be expressed as one-fourth (D) since it is the closest option.