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Activity 4: Taking Chances with Events A or B

1. Problem 1:

A bowl contains 15 chips numbered 1 to 15. If a chip is drawn randomly from the bowl, what is the probability that it is:

• a multiple of 3
• a number divisible by 5
• divisible by both 3 and 5

Solution:

• Multiples of 3: {3, 6, 9, 12, 15} → There are 5 such numbers.
  $P(\text{Multiple of 3}) = \frac{5}{15} = \frac{1}{3}$
• Multiples of 5: {5, 10, 15} → There are 3 such numbers.
  $P(\text{Multiple of 5}) = \frac{3}{15} = \frac{1}{5}$
• Multiples of both 3 and 5 (i.e., divisible by 15): {15} → Only 1 such number.
  $P(\text{Multiple of both 3 and 5}) = \frac{1}{15}$
  Using the inclusion-exclusion principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ $P(\text{Multiple of 3 or 5}) = \frac{5}{15} + \frac{3}{15} - \frac{1}{15} = \frac{7}{15}$

2. Problem 2:

One puts 44 marbles in a box: 14 are red, 12 are blue, and 18 are yellow. If one picks one marble at random, what is the probability the marble is red or yellow?

Solution:
The total number of marbles is: $44 = 14 + 12 + 18$ The probability of picking a red marble: $P(\text{Red}) = \frac{14}{44}$ The probability of picking a yellow marble: $P(\text{Yellow}) = \frac{18}{44}$ Since these events are mutually exclusive, we can add the probabilities: $P(\text{Red or Yellow}) = \frac{14}{44} + \frac{18}{44} = \frac{32}{44} = \frac{8}{11}$

3. Problem 3:

Out of 5200 households surveyed, 2107 had a dog, 807 had a cat, and 310 had both a dog and a cat. What is the probability that a randomly selected household has a dog or a cat?

Solution:
Using the inclusion-exclusion principle: $P(\text{Dog or Cat}) = P(\text{Dog}) + P(\text{Cat}) - P(\text{Dog and Cat})$ $P(\text{Dog}) = \frac{2107}{5200}, \quad P(\text{Cat}) = \frac{807}{5200}, \quad P(\text{Dog and Cat}) = \frac{310}{5200}$ $P(\text{Dog or Cat}) = \frac{2107}{5200} + \frac{807}{5200} - \frac{310}{5200}$ $= \frac{2604}{5200} = \frac{501}{1000} = 0.501$

Would you like detailed explanations of the Venn diagrams or a deeper analysis of mutual exclusivity in these events?

Five Related Questions

1. How do you visualize probabilities using Venn diagrams effectively?
2. What distinguishes mutually exclusive events from independent events?
3. How does the inclusion-exclusion principle apply in real-life situations?
4. How do probabilities change when sample spaces vary?
5. What are practical applications of probabilities in surveys?

Tip:

When calculating "or" probabilities, always check whether the events overlap to apply the inclusion-exclusion principle properly.