5. Consider an experiment in which a fair coin is tossed once and a balanced dice is rolled once. a) Describe the sample space for this experiment. b) Find the probability that a head will be obtained on the coin and an odd number will be obtained on the dice? 6. A hat contains five pieces of paper, two of which have the number 2 written on them and three of which have the number 3 written on them. Now suppose you randomly select one piece of paper from the hat. Find the cumulative distribution function (cdf) for the number X that is showing on the paper. 7. Suppose ten runners enter a race and the first three runners to cross the finish line receive the same prize. In how many ways can the prizes be distributed?
Added by David W.
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Sample space = {HH, HT, H1, H2, H3, H4, H5, H6, TH, TT, T1, T2, T3, T4, T5, T6} Show more…
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A game consists of first rolling an ordinary six-sided die once and then tossing an unbiased coin once. The score, which consists of adding the number of dots showing on the die, and the number of heads showing on the coin (0 or 1), is a random variable, say X. a) List the possible values of X, and write its PMF in the form of a table. b) Draw a graph of the PMF. c) What is the CDF of X? Write it in the form of a table. d) Draw a graph of the CDF of X. e) Find P[X > 3]. f) Find the probability that the score is an odd integer.
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Consider the following discrete random variables: (a) X has a Bernoulli distribution, with P[X = 1] = 2/3, P[X = 0] = 1/3 (an unfair coin toss). (b) X takes the values 1,2,3,4,5, and 6, with each outcome equally likely, P(X = j) = 1/6 for j = 1,2,3,4,5,6 (the result of rolling a fair, six-sided die). (c) X takes the value 1,2,3,4, and 6, with P(X = j) = 1/6 for j = 1,2,3,4, and P(X = 6) = 1/3 (the result of rolling a loaded die that will never land on 5). For each of the above examples: (a) Graph the probability mass function; (b) Write down the equation for the cdf Fx. Graph Fx. What is the support of the distribution of X? What is P(X < 1)? What is P(X < 3)? What is P(X ∈ {1,3,5})? What is P(1 < X ≤ 3)? (h) What is P(1 ≤ X ≤ 3)? What is E(X)?
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One way to solve a probability problem is to repeat the experiment many times, keeping track of the results. Then the probability can be approximated using the basic definition of the probability of an event $E : P(E)=n(E) / n(S),$ where $E$ occurs (E) times out of $n(S)$ trials of an experiment. This is called the Monte Carlo method of finding probabilities. If physically repeating the experiment is too tedious, it may be simulated using a random-number generator, availators. To most computers and scientific or graphing calculators. To simulate a coin toss or the roll of a die on the TI-84 Plus, change the setting to fixed decimal mode with 0 digits displayed, and enter rand or rand $^{\star} 6+.5,$ respectively. For a coin toss, interpret 0 as a head and 1 as a tail. In either case, the ENIER key can be pressed repeatedly to perform multiple simulations. Suppose a coin is tossed 5 times. Use the Monte Carlo method with at least 50 repetitions to approximate the following probabilities. a. P(exactly 4 heads) b. P(2 heads and 3 tails)
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