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Preparing for the next class

In the next in-class activity, you will need to calculate conditional probabilities,  probabilities relating to independent events, and probabilities relating to mutually exclusive events.

In previous discussions of probability, we might have given you information on Rafael’s  activities and then asked, “What is the probability that Rafael rides his bike?”

In this activity, we will calculate probabilities in the presence of additional information. For example, “Given that it is raining, what is the probability that Rafael rides his bike?”

Let’s try.

Question 1

Diamond has four phone cases. Two phone cases have a pop socket while the other  two do not. Every morning, Diamond closes her eyes and randomly selects one  phone case to use for the day.

  1. What is the probability that Diamond selects a phone case with a pop  socket?
  2. One morning, before Diamond wakes up, her little sister borrows one of the  cases with a pop socket. Given that Diamond’s sister has taken one of the  pop socket phone cases, what is the probability that Diamond randomly  selects a phone case with a pop socket?
    Hint: There is one phone case with a pop socket in the three remaining phone  cases.We call the probability in Part B a conditional probability because we calculated  Diamond’s probability of selecting a case with a pop socket conditional on the fact  that her sister had taken one of her cases. We write this as:
    P(Diamond selects case with pop socket GIVEN her sister took one case with pop socket)
    The probability of Diamond selecting a phone case with a pop socket is impacted by  her sister borrowing one of the phone cases. In this example, one event occurring impacts the probability of another event occurring. However, this is not always the  case.

    Sometimes, there are pairs of events for which one event has no effect on the  probability of another event occurring. When this is the case, we say the events are  independent. More formally:
    Events A and B are independent if [latex]P(A~GIVEN~B) = P(A)[/latex]

  3. Determine whether the following is true or false:P(Diamond selects case with pop socket GIVEN her sister took one case  with pop socket) = P(Diamond selects case with pop socket)

    Hint: Are your answers to Parts A and B the same?

  4. Determine whether the following statement is true or false:Whether Diamond selects a phone case with a pop socket is independent of  her sister borrowing one of her pop socket phone cases.

Question 2

On any given day, the probability of Rafael checking his email is 0.9. The weather  does not affect the probability of him checking his email.

  1. Do you think rainy weather and Rafael checking his email are independent  events?
    Hint: The problem tells us the weather does not affect the probability of him checking  his email.
  2. Calculate the probability of Rafael checking his email conditional on the fact  that it is raining. That is, calculate: P(Rafael checks his email GIVEN  Raining).
    Hint: The probability of checking email on a rainy day is the same as the probability  of checking email in general.

Question 3

For each part, determine whether the two events are independent.

  1. Rafael avoids riding his bike in the rain. Is biking to work independent of  rain?
    Hint: If the rain affects Rafael’s probability of biking, then the two events are not  independent.
  2. On any given day, the probability that Hiroki wears sneakers is 85%. On  Fridays, the probability that Hiroki wears sneakers is 85%. Is wearing  sneakers independent of whether it’s a Friday?

Recall that two events are mutually exclusive if the probability of both events happening  at the same time is zero. For example, consider flipping a coin. It can land heads up or  heads down, but it cannot be both heads up and heads down simultaneously. Thus,  heads and tails are mutually-exclusive events.

Question 4

For each part, determine if the pair of events are mutually exclusive.

  1. The weather is rainy and Rafael checks his email
    Hint: Is P(Rafael checks his email and it’s raining) = 0?
  2. Eating a steak and being veganHint: A person is vegan if they do not eat meat.
  3. Wearing shoes and being barefoot
  4. Wearing a dress and having pockets