First, "complete" the table by counting up the totals in each category.
Total number of students that can swim:
215 + 269 + 293 = 777
Total number of students that cannot swim:
85 + 31 + 7 = 123
Total number of Year 4 students:
215 + 85 = 300
Total number of Year 7 students:
269 + 31 = 300
Total number of Year 10 students:
293 + 7 = 300
Total number of students:
777 + 123 = 300 + 300 + 300 = 900
Then the probability that a randomly selected student fall in a given category is equal to (the number of students in that category) divided by (the total number of students at the school).
(a) can swim:
777/900 = 259/300 ≈ 0.86
(b) cannot swim:
123/900 = 41/300 ≈ 0.14
(notice that this is also equal to 1 minus the probability that a student can swim)
(c) from year 7:
300/900 = 1/3 ≈ 0.33
(d) from year 7 AND cannot swim:
31/900 ≈ 0.034
(e) from year 7 OR cannot swim:
(300 + 123 - 31)/900 = 98/225 ≈ 0.44
This follows from the inclusion/exclusion principle.
P(A or B) = P(A) + P(B) - P(A and B)
(f) cannot swim, given from year 7 OR year 10:
(38/900) / (600/900) = 38/600 = 19/300 ≈ 0.063
By definition of conditional probability,
P(cannot swim | year 7 OR year 10) = P(cannot swim AND (year 7 OR year 10)) / P(year 7 OR year 10)
There are 600 students in either year 7 or 10, and according to the table, 31 + 7 = 38 of them cannot swim.
(g) from year 4, given that they cannot swim:
(85/900) / (123/900) = 85/123 ≈ 0.69
Again, by definition,
P(year 4 | cannot swim) = P(year 4 AND cannot swim) / P(cannot swim)
