Probability-Event relations

Front

In a class of 50 students, 30 students use an iPhone (Event A) and 20 students use a Mac (Event B). There are 12 students who use both an iPhone and a Mac. If one student is selected at random, what is the probability that they use an iPhone OR a Mac?

Back

Solution:

1. Identify the probabilities:

   • P(A) = 30/50 = 0.60

   • P(B) = 20/50 = 0.40

   • P(A \cup B) = 12/50 = 0.24

2. Apply the Additive Rule:

   P(A \cup B) = P(A) + P(B) - P(A \cap B)
   = 0.60 + 0.40 - 0.24 = 0.76

   Final Answer: There is a 76% chance the student uses at least one of the two devices.

Front

In the same class, 10 students use only Android phones (Event C). We know that no student uses both an iPhone and an Android. If you pick a student at random, what is the probability they use an iPhone (Event A) OR an Android (Event C)?

Back

Solution:

1. Identify the probabilities:

   • P(A) = 30/50 = 0.60

   • P(C) = 10/50 = 0.20

2. Determine Relationship: Since no student uses both, these events are Mutually Exclusive ((A \cap C) = 0).

3. Apply the Rule for Mutually Exclusive Events:

   P(A \cup C) = P(A) + P(C)

   = 0.60 + 0.20 = 0.80

   Final Answer: There is an 80% chance the student uses one of these two phone types.

Front

Using the results from Question 1 (where the probability of using an iPhone or a Mac was 0.76), what is the probability that a randomly selected student uses neither an iPhone nor a Mac?

Back

Solution:

1. Identify the Event: The event "neither" is the Complement of the Union.

2. Apply the Complement Rule:

   P(Neither) = 1 - P(A \cup B)

   P(Neither) = 1 - 0.76 = 0.24

   Final Answer: There is a 24% chance the student uses neither device.