5.8 Exercises

  1. Simple Probabilities (**): Calculate the following probabilities:

    1. Drawing an ace from a deck of cards?
    2. Getting a number divisible by 3 after rolling a die?
    3. Sum of two dice being equal to 7?
    4. Getting at least one head after flipping a coin twice?

  2. Probability of a Union I (*): Consider the two events A and B which are mutually exclusive. The probabilities associated with those two events are Pr(A)=0.23 and Pr(B)=0.47. What is the probability of A or B occurring? What is the probability that neither occurs?

  3. Probability of a Union II (*): Consider the probability associated with the two events A and B, i.e., Pr(A)=0.45 and Pr(B)=0.3. The probability that both of those events occur at the same time, i.e., \(Pr(A\cap B)\) is 0.2. What is the probability of the union?

  4. Party (**): Assume that 52% of the population are Republicans and 48% of the population are Democrats. On a particular issue, 64% of Republicans are in favor and 52% of Democrats are in favor. If you randomly pick a person who is in favor of the issue, what is the probability that the person is a Democrat?

  5. Smoke Detector (**): Smoke and fire detectors are essential to save lives. Unfortunately, there are many false fire alarms. Suppose that the probability of an actual fire happening is very low at 5%. Smoke detectors are extremely good at detecting an actual fire, i.e., given a fire, there is a 99% probability that the smoke alarm detects it. If there is no fire, there is a 10% probability that the fire alarm sounds. Suppose that you hear a fire alarm, what is the probability that there is a fire?

  6. Independence (**): Suppose you are rolling a die. Consider the events A (“rolling an even number”) and B (“rolling a number less than four”). Are A and B independent?