Questions and Answers of Probability With Applications

Suppose P(ABC) = 0.05, P(AB) = 0.15, P(AC) = 0.2, P(BC) = 0.25, P(A) = P(B) = P(C) = 0.5. For each of the events given next, write the event using set notation in terms of A, B, and C, and compute
An exam had three questions. One-fifth of the students answered the first question correctly; one-fourth answered the second question correctly; and one-third answered the third question correctly.
The odds in favor of an event is the ratio of the probability that the event occurs to the probability that it will not occur. For example, the odds that you were born on a Friday, assuming birth
Let A, B, C be three events. Find expressions for the events:(a) At least one of the events occurs.(b) Only B occurs.(c) At most one of the events occurs.(d) All of the events occur.(e) None of the
The functionis a probability function for some choice of c. Find c. 3k = C- P(k) for k = 0, 1, 2, ..., P(k) = c k!
Let(a) Show that Q is a probability function. That is, show that the terms are non-negative and sum to 1.(b) Let X be a random variable such that P(X = k) = Q(k), for k = 0, 1, 2, . . . . Find P(X
A tetrahedron dice is four-sided and labeled with 1, 2, 3, and 4. When rolled it lands on the base of a pyramid and the number rolled is the number on the base. In five rolls, what is the probability
Two dice are rolled. Let X be the maximum number obtained. (Thus, if 1 and 2 are rolled, X = 2; if 5 and 5 are rolled, X = 5.) Assume that all 36 elements of the sample space are equally likely. Find
Four coins are tossed. Let A be the event that the first two coins comes up heads. Let B be the event that the number of heads is odd. Assume that all 16 elements of the sample space are equally
See the assignment of probabilities to the Venn diagram in Figure 1.4. Find the following:(a) P(No events occur).(b) P(Exactly one event occurs).(c) P(Exactly two events occur).(d) P(Exactly three
Let A, B, C, be three events. At least one event always occurs. But it never happens that exactly one event occurs. Nor does it ever happen that all three events occur. If P(AB) = 0.10 and P(AC) =
Suppose X is a random variable that takes values on {0, 0.01, 0.02, . . . , 0.99, 1}. If each outcome is equally likely, find(a) P(X ≤ 0.33).(b) P(0.55 ≤ X ≤ .66).
Suppose X is a random variable that takes values on all positive integers. Let A = {2 ≤ X ≤ 4} and B = {X ≥ 4}. Describe the events (i) Ac(ii) Bc(iii) AB(iv) A ∪ B
Suppose P(A ∪ B) = 0.6 and P(A ∪ Bc) = 0.8. Find P(A).
Suppose A and B are mutually exclusive, with P(A) = 0.30 and P(B) = 0.60. Find the probability that(a) At least one of the two events occurs(b) Both of the events occur(c) Neither event occurs(d)
Suppose P(A) = 0.40, P(B) = 0.60, and P(A or B) = 0.80. Find(a) P(neither A nor B occur).(b) P(AB).(c) P(one of the two events occurs, and the other does not).
Suppose k numbers are chosen from {1, . . . , n}, where k < n, sampling without replacement. All outcomes are equally likely. What is the probability that the numbers chosen are in increasing
Amy is picking her fall term classes. She needs to fill three time slots, and there are 20 distinct courses to choose from, including probability 101, 102, and 103. She will pick her classes at
Suppose you throw five dice and all outcomes are equally likely.(a) What is the probability that all dice are the same?(b) What is the probability of getting at least one 4?(c) What is the
Suppose that license plates can be two, three, four, or five letters long, taken from the alphabets A to Z. All letters are possible, including repeats. A license plate is chosen at random in such a
Let P1 and P2 be two probability functions on Ω. Define a new function P such that P(A) = (P1(A)+P2(A))/2. Show the P is a probability function.
A fair coin is flipped six times. What is the probability that the first two flips are heads and the last two flips are tails? Use the multiplication principle.
A club has 10 members and is choosing a president, vice-president, and treasurer. All selections are equally likely.(a) What is the probability that Tom is selected president?(b) What is the
Let P be a probability function on Ω = {a, b} such that P(a) = p and P(b) = 1 − p for 0 ≤ p ≤ 1. Let Q be a function on Ω defined by Q(ω) = [P(ω)]2. For what value(s) of p will Q be a valid
A random experiment has three possible outcomes a, b, and c, with P(a) = p, P(b) = p2, and P(c) = p. What choice(s) of p makes this a valid probability model?
A sample space has four elements ω1, . . . , ω4 such that ω1 is twice as likely as ω2, which is three times as likely as ω3, which is four times as likely as ω4. Find the probability function.
A couple plans to continue having children until they have a girl or until they have six children, whichever comes first. Describe a sample space and a reasonable random variable for this random
A bag contains r red and b blue balls. You reach into the bag and take k balls. Let R be the number of red balls you take. Let B be the number of blue balls. Express the following events in terms of
In two dice rolls, let X be the outcome of the first die, and Y the outcome of the second die. Then X + Y is the sum of the two dice. Describe the following events in terms of simple outcomes of the
Bored one day, you decide to play the video game Angry Birds until you win.Every time you lose, you start over. Consider the probability that you win in less than 1000 tries.(i) The random
Your friend was sick and unable to make today’s class. Explain to your friend, using your own words, the meaning of the terms: (i) Random experiment, (ii) Sample space, (iii) Event,
In Angel’s garden, there is a 3% chance that a tomato will be bad. Angel harvests 100 tomatoes and wants to know the probability that at most five tomatoes are bad.(i) The random experiment.(ii)
Roll four dice. Consider the probability of getting all fives.(i) The random experiment, (ii) Sample space, (iii) Event, and (iv) Random variable. Express the probability in
A pizza shop offers three toppings: pineapple, peppers, and pepperoni. A pizza can have 0, 1, 2, or 3 toppings. Consider the probability that a random customer asks for two toppings.(i) The random

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