The Problem:

We have looked at the probability that two non-mutually exclusive events A or B can happen in this learning experience.

We will take it one step further and look at the probability that event A or B or C will happen.

The formula is:
P[A or B or C] = P[A] + P[B] + P[C] − P[A ∩ B] - P[B ∩ C] - P[A ∩ C] + P[A ∩ B ∩ C]
Use the formula to answer the following problem.

The numbers 1 to 40 are all written on slips of paper and placed into a bag. Find the probability to 3 decimal places of drawing a composite number above 20 or multiples of 4 or multiples of 3.

1 mark for substituting correct values on the first line
3 marks for showing work
1 mark for correct final probability

To find the probability of drawing a composite number above 20 or multiples of 4 or multiples of 3 from the given numbers, we can use the formula:

P[A or B or C] = P[A] + P[B] + P[C] - P[A ∩ B] - P[B ∩ C] - P[A ∩ C] + P[A ∩ B ∩ C]

Let's break down the problem and calculate each probability step by step:

A: Composite numbers above 20
To find P[A], we need to determine the number of composite numbers above 20 (excluding 20 itself) and divide it by the total number of possible outcomes (40 numbers in the bag).

Total composite numbers above 20: 23, 24, 25, ..., 40 (total of 18 numbers)
Total possible outcomes: 40

P[A] = 18/40 = 9/20

B: Multiples of 4
To find P[B], we need to determine the number of multiples of 4 and divide it by the total number of possible outcomes (40 numbers in the bag).

Total multiples of 4: 4, 8, 12, ..., 40 (total of 10 numbers)
Total possible outcomes: 40

P[B] = 10/40 = 1/4

C: Multiples of 3
To find P[C], we need to determine the number of multiples of 3 and divide it by the total number of possible outcomes (40 numbers in the bag).

Total multiples of 3: 3, 6, 9, ..., 39 (total of 13 numbers)
Total possible outcomes: 40

P[C] = 13/40

A ∩ B: Composite numbers above 20 and multiples of 4
To find P[A ∩ B], we need to determine the number of composite numbers above 20 that are also multiples of 4 (excluding 20 itself) and divide it by the total number of possible outcomes (40 numbers in the bag).

Total composite numbers above 20 and multiples of 4: 24, 28, 32, 36, 40 (total of 5 numbers)
Total possible outcomes: 40

P[A ∩ B] = 5/40 = 1/8

B ∩ C: Multiples of 4 and multiples of 3
To find P[B ∩ C], we need to determine the number of multiples of 4 that are also multiples of 3 and divide it by the total number of possible outcomes (40 numbers in the bag).

Total multiples of 4 and multiples of 3: 12, 24, 36 (total of 3 numbers)
Total possible outcomes: 40

P[B ∩ C] = 3/40

A ∩ C: Composite numbers above 20 and multiples of 3
To find P[A ∩ C], we need to determine the number of composite numbers above 20 that are also multiples of 3 (excluding 20 itself) and divide it by the total number of possible outcomes (40 numbers in the bag).

Total composite numbers above 20 and multiples of 3: 24, 27, 30, 33, 36, 39 (total of 6 numbers)
Total possible outcomes: 40

P[A ∩ C] = 6/40 = 3/20

A ∩ B ∩ C: Composite numbers above 20, multiples of 4, and multiples of 3
To find P[A ∩ B ∩ C], we need to determine the number of composite numbers above 20 that are multiples of both 4 and 3 (excluding 20 itself) and divide it by the total number of possible outcomes (40 numbers in the bag).

Total composite numbers above 20, multiples of 4, and multiples of 3: 24, 36 (total of 2 numbers)
Total possible outcomes: 40

P[A ∩ B ∩ C] = 2/40 = 1/20

Now, substitute the values we found into the formula:

P[A or B or C] = P[A] + P[B] + P[C] - P[A ∩ B] - P[B ∩ C] - P[A ∩ C] + P[A ∩ B ∩ C]
P[A or B or C] = 9/20 + 1/4 + 13/40 - 1/8 - 3/40 - 3/20 + 1/20

Simplify the expression:

P[A or B or C] = 18/40 + 10/40 + 13/40 - 5/40 - 3/40 - 6/40 + 2/40
P[A or B or C] = 29/40

Therefore, the probability of drawing a composite number above 20 or multiples of 4 or multiples of 3 from the given numbers is 29/40, which can be approximated to 0.725 when rounded to three decimal places.