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Describe the sample space \(S\) of the experiment and list the elements of the given event. (Assume that the coins are distinguishable and that what is observed are the faces or numbers that face up.) HINT [See Examples 1-3.] Two coins are tossed: the result is at most one tail

Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, three green ones, two white ones, and one purple one. She grabs five of them. Find the probabilities of the following events, expressing each as a fraction in lowest terms. HINT [See Example 1.] She does not have all the green ones.

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the coins and dice are distinguishable and fair, and that what is observed are the faces or numbers uppermost. Three coins are tossed; the result is at most one head.

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the coins and dice are distinguishable and fair, and that what is observed are the faces or numbers uppermost. Two dice are rolled; the numbers add to 9 .

You are given a transition matrix \(P\) and initial distribution vector \(v\). Find (a) the two-step transition matrix and (b) the distribution vectors after one, two, and three steps. $$ P=\left[\begin{array}{cc} 1 / 2 & 1 / 2 \\ 1 & 0 \end{array}\right], v=\left[\begin{array}{ll} 2 / 3 & 1 / 3 \end{array}\right] $$

Describe the sample space \(S\) of the experiment and list the elements of the given event. (Assume that the coins are distinguishable and that what is observed are the faces or numbers that face up.) HINT [See Examples 1-3.] A sequence of two different digits is randomly chosen from the digits \(0-4\); the first digit is twice the second.

If two indistinguishable dice are rolled, what is the probability of the event \(\\{(4,4),(2,3)\\} ?\) What is the corresponding event for a pair of distinguishable dice? HINT [See Example 2.]

A die is weighted in such a way that each of 2,4, and 6 is twice as likely to come up as each of 1,3, and \(5 .\) Find the probability distribution. What is the probability of rolling less than 4 ?

In exercise, you are asked to calculate the probability of being dealt various poker hands. (Recall that a poker player is dealt 5 cards at random from a standard deck of 52.) Express each of your answers as a decimal rounded to four decimal places, unless otherwise stated. Two cards with one denomination, two with another, and one with a third.

Another die is weighted in such a way that each of 1 and 2 is three times as likely to come up as each of the other numbers. Find the probability distribution. What is the probability of rolling an even number?