Combination Practice Question (Please Provide an explanation for each scenario!)

Question

1.A survey of 1000 television viewers conducted by a local television station produced the following data:

• 40% watch the news at 12:00

• 60% watch the news at 18:00

• 50% watch the news at 23:00

• 25% watch the news at 12:00 and at 18:00

• 20%watch the news at 12:00 and 23:00

• 20% watch the news at 18:00 and 23:00

• 10% watch all three news broadcasts

a) What percent of those surveyed watch at least one of these programs?

b) What percent watch none of these news broadcasts?

c) What percent view the news at 12:00 and at 18:00, but not at 23:00?

d) What percent view only one of these three shows?

e) What percent view exactly two of these shows?

Please provide an explanation for each (including question if can), it would be extremely helpful. Thank you

Answer 1

I will show you two ways to solve these questions. One relies on the inclusion/exclusion property, etc. The other just calculates the probability of each of the eight possible combinations of watching/not watching.at the three times.

Method 1.

a) inclusion/exclusion. singlets - doublets + triplets (if quadruplets, would subtract; if quintuplets, add back in)

40% + 60% + 50% - 25% - 20% - 20% + 10% = 95%

b) 100% - a) = 100% -95% = 5%

c) 12 and18 but not 23 is number watching 12 and 18 - number watching all 3 = 25% - 10% = 15%

d) Number watching only one show = number watching - number watching each of the pairs + 2 * the number watching all three (the number watching all three would have also been counted 3 times before in each of the doublets; to remove only once from the total, add back in 2)

95% - 25% - 20% - 20% + 2 * 10% = 50%

e) Number watching exactly two shows = number watching each of the pairs of shows - 3 * the number watching all three shows, as these would have been counted in each pair.

25% + 20%+ 20% - 3 * 10% = 35%

(Note that the answer to d + the answer to e + 10%, the number watching all 3 = 95%, our answer to a)

Method 2: first solve for the eight groups A = watch at 12:00; B = watch at 18:00; C = watch at 23:00

ABC = 10%

ABC' = AB - ABC = 25% - 10% = 15%

AB'C = AC - ABC = 20% - 10% = 10%

A'BC = BC - ABC = 20% - 10% = 10%

AB'C' = A - A(B ∪ C) = A - AB - AC + ABC = 40% - 25% - 20% + 10% = 5%

A'BC' = B - B(A ∪ C) = B - AB - BC + ABC = 60% - 25% - 20% + 10% = 25%

A'B'C = C - C(A ∪ B) = C - AC - BC + ABC = 50% - 20% - 20% + 10% = 20%

A'B'C' = 1 - the answers above = 1 - 10% - 15% - 10% - 10% - 5% - 25% - 20% = 5%

Then, a) = either the sum of the seven individual groups or 1 - A'B'C' = 1 - 5% = 95%

b) A'B'C' = 5%

c) ABC' = 15%

d) AB'C' + A'BC' + A'B'C = 5% + 25% + 20% = 50%

e) ABC' + AB'C + A'BC = 15%+ 10% + 10% = 35%