91Ó°ÊÓ

List three methods of assigning probabilities.

You need to know the number of different arrangements possible for five distinct letters. You decide to use the permutations rule, but your friend tells you to use \(5 !\). Who is correct? Explain.

Given \(P(A)=0.2\) and \(P(B)=0.4\) : (a) If \(A\) and \(B\) are independent events, compute \(P(A\) and \(B)\). (b) If \(P(A \mid B)=0.1\), compute \(P(A\) and \(B)\).

What is the probability that a day of the week selected at random will be a Wednesday?

Rules of Probability Given \(P\left(A^{c}\right)=0.8, P(B)=0.3\), \(P(B \mid A)=0.2:\) (a) Compute \(P(A\) and \(B)\). (b) Compute \(P(A\) or \(B)\).

Consider a family with 3 children. Assume the probability that one child is a boy is \(0.5\) and the probability that one child is a girl is also \(0.5\), and that the events "boy" and "girl" are independent. (a) List the equally likely events for the gender of the 3 children, from oldest to youngest. (b) What is the probability that all 3 children are male? Notice that the complement of the event "all three children are male" is "at least one of the children is female." Use this information to compute the probability that at least one child is female.

(a) Explain why \(-0.41\) cannot be the probability of some event. (b) Explain why \(1.21\) cannot be the probability of some event. (c) Explain why \(120 \%\) cannot be the probability of some event. (d) Can the number \(0.56\) be the probability of an event? Explain.

Can you raise one eyebrow at a time? Use the students in your statistics class (or a group of friends) to estimate the percentage of people who can raise one eyebrow at a time. How can your result be thought of as an estimate for the probability that a person chosen at random can raise one eyebrow at a time? Comment: National statistics indicate that about \(30 \%\) of Americans can raise one eyebrow at a time (see source in Problem 13).

You draw two cards from a standard deck of 52 cards without replacing the first one before drawing the second. (a) Are the outcomes on the two cards independent? Why? (b) Find \(P(3\) on 1 st card and 10 on 2 nd ). (c) Find \(P(10\) on 1 st card and 3 on 2 nd ). (d) Find the probability of drawing a 10 and a 3 in either order.

You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck. (a) Are the outcomes on the two cards independent? Why? (b) Find \(P(3\) on 1 st card and 10 on 2 nd). (c) Find \(P(10\) on 1 st card and 3 on 2 nd). (d) Find the probability of drawing a 10 and a 3 in either order.