Of one hundred students surveyed: 70 like
basketball, 52 like football, 61 like baseball, 29 like
baseball and football, 39 like football and
basketball, 42 like basketball and baseball, 27 like
all three sports. If a student is chosen at random,
what is the probability that she/he likes no more
than one sport
A - basketball
B - Football
C - baseball
Numer(A or B or C) = number(A) + number(B) + number(C) - number(A and B) - number(A and C) - number(B and C) + number(A and B and C)
100 = 70 +52 +61 -39 -29 - 42 + 27
Data is valid.
make Venn diagrams, 3 intersecting circles , call them A, B, and C
put 27 in the intersection of all 3
39 like football and basketball, but 27 of those have already been counted, so put 12 in the intersection of those two outside the 27
42 like baseball and basketball, but 27 of those have already been counted, so put 15 in the region of A and C outside the 27.
29 like baseball and football but 27 are already counted, so put 2 in the intersection of B and C, ouside the 27
now look at A, I count up 54 so far as liking Basketball, but we were told that 70 like basketball, which means we have to put 16 in the region of A not intersected with any of the others.
Continue in this way to fill in the other parts.
Now count up all the numbers of A, B, and C that do not intersect with any of the others.
Put that number of 100 to get your probability.
To find the probability that a randomly chosen student likes no more than one sport, we need to count the number of students who like either no sport or only one sport.
Let's break down the given information to calculate the probability step by step:
1. Total number of students surveyed: 100.
2. Number of students who like basketball: 70.
(Let's represent this as |Basketball|)
3. Number of students who like football: 52.
(Represented as |Football|)
4. Number of students who like baseball: 61.
(Represented as |Baseball|)
5. Number of students who like baseball and football: 29.
(Represented as |Football ∩ Baseball|)
6. Number of students who like basketball and football: 39.
(This is represented as |Basketball ∩ Football|)
7. Number of students who like basketball and baseball: 42.
(Represented as |Basketball ∩ Baseball|)
8. Number of students who like all three sports: 27.
(Represented as |Basketball ∩ Football ∩ Baseball|)
Now, to calculate the number of students who like no more than one sport, we need to add up a few sets:
9. Number of students who like only basketball:
(|Basketball| - |Basketball ∩ Football| - |Basketball ∩ Baseball| + |Basketball ∩ Football ∩ Baseball|)
10. Number of students who like only football:
(|Football| - |Basketball ∩ Football| - |Football ∩ Baseball| + |Basketball ∩ Football ∩ Baseball|)
11. Number of students who like only baseball:
(|Baseball| - |Basketball ∩ Baseball| - |Football ∩ Baseball| + |Basketball ∩ Football ∩ Baseball|)
12. Number of students who like no sport:
(Total number of students surveyed - Sum of students who like only a specific sport - |Basketball ∩ Football ∩ Baseball|)
Finally, we can calculate the probability by dividing the total number of students who like no more than one sport by the total number of students surveyed.
Probability = (Number of students who like no more than one sport) / (Total number of students surveyed)