Assignment 2-3 Probability and Probability Distributions

Assignment 2-3 Probability and Probability Distributions -...

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1. Consider the experiment of tossing a fair coin four times. The coin has two possible outcomes, heads or tails. a. List the sample space for the outcomes that could happen when tossing the coin four times. For example, if all four coin tosses produced heads, then the outcome would be HHHH. b. If each outcome is equally likely, what is the probability that all four coin tosses result in heads? Notice that the complement of “all four heads” is “at least one tail.” Using this information, compute the probability that there will be at least one tail out of the four coin tosses. 1.a) HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THTH, TTHH, THHT, HTTT, THTT, TTHT, TTTH, TTTT 1.b) P ( all 4 heads ) = 1 16 P ( at least 1 tail ) = 1 P ( all 4 heads ) = 1 1 16 = 15 16 2. Suppose you roll a single fair die and note the number rolled. a. What is the sample space for a single roll of a fair die? Are the outcomes equally likely? b. Assign probabilities to the outcomes in the sample space found in part (a). Do these probabilities add up to 1? Should they add up to 1? Why? c. What is the probability of getting a number less than 4 on a single roll? d. What is the probability of getting a 1 or a 2 on a single roll? 2.a) The sample space for a single roll of a fair die is {1, 2, 3, 4, 5, 6} and all outcomes are equally likely to occur. 2.b) P ( 1 ) = 1 6 P ( 2 ) = 1 6 P ( 3 ) = 1 6 P ( 4 ) = 1 6 P ( 5 ) = 1 6 P ( 6 ) = 1 6 P ( 1 ) + P ( 2 ) + P ( 3 ) + P ( 4 ) + P ( 5 ) + P ( 6 ) = 1 6 + 1 6 + 1 6 + 1 6 + 1 6 + 1 6 = 6 6 = 1 The probabilities of all possible outcomes must be 1 because one of the outcomes must be obtained 2.c) P ( numberlessthan 4 ) = P ( 1 ) + P ( 2 ) + P ( 3 ) = 1 6 + 1 6 + 1 6 = 3 6 = 1 2 2.d)Since the events in question are mutually exclusive then P ( 1 2 ) = P ( 1 ) + P ( 2 ) = 1 6 + 1 6 = 2 6 = 1 3
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3. Suppose we are interested in studying movie ratings where movies get rated on a five star scale. One star means the critic thought the movie was horrible, and five stars means the critic thought it was one of the best movies of the year. Here is a frequency table for all the movies rated by this critic for the year: a. Using this information, if we chose a movie from this group at random, what is the probability that the movie received a: 1 star rating? 2 star rating? 3 star rating? 4 star rating? 5 star rating? b. Do the probabilities from part (a) add up to 1? Why should they? What is the sample space in this problem? 3.a) P ( 1 star ) = 28 852 P ( 2 star ) = 123 852 P ( 3 star ) = 356 852 P ( 4 star ) = 289 852 P ( 5 star ) = 56 852 3.b) The probabilities from part (a) add to 1 because from the probabilities of all possible outcomes, one of the outcomes must be obtained, and it is the sum of all the probabilities of the sample space.
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