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a. Use the line of random numbers below to simulate flipping a coin 20 times. Use the digits \(0,1,2,3,4\) to represent heads and the digits \(5 .\) 6\. \(7,8,9\) to represent tails. $$ \begin{array}{llll} 11164 & 36318 & 75061 & 37674 \end{array} $$ b. Based on these 20 trials, what is the simulated probability of getting heads? How does this compare with the theoretical probability of getting heads? c. Suppose you repeated your simulation 1000 times and used the simulation to find the simulated probability of getting heads. How would the simulated probability compare with the theoretical probability of getting heads?

Imagine flipping a fair coin many times. Explain what should happen to the proportion of heads as the number of coin flips increases.

Consider two pairs of grandparents. The first pair has 4 grandchildren, and the second pair has 32 grandchildren. Which of the two pairs is more likely to have between \(40 \%\) and \(60 \%\) boys as grandchildren, assuming that boys and girls are equally likely as children? Why?

Some estimates say that \(10 \%\) of the population is left-handed. We wish to design a simulation to find an empirical probability that if five babies are born on a single day, one or more will be left-handed. Suppose we decide that the even digits \((0,2,4 .\) 6,8 ) will represent left-handed babies and the odd digits will represent right-handed babies. Explain what is wrong with the stated simulation method, and provide a correct method.

a. Fxplain how you could use a random number table to simulate rolling a fair six-sided die 20 times. Assume you wish to find the probability of rolling a I. Then report a line or two of the random number table (or numbers generated by a computer or calculator) and the values that were obtained from it. b. Report the empirical probubility of rolling a 1 from part (u), and compare it with the theoretical probability of rolling a \(1 .\)

Let \(H\) stand for heads and let \(T\) stand for tails in an experiment where a fair coin is flipped twice. Assume that the four outcomes listed are equally likely outeomes: $$ \mathrm{HH}, \mathrm{HT}, \mathrm{TH}, \mathrm{TT} $$ What are the probabilities of getting the following: a. 0 heads b. Exactly 1 head c. Lixactly 2 heads d. At least one 1 head e. Not more than 2 heads

A bag contains a number of colored cubes: 10 red, 5 white, 20 blue, and 15 black. One cube is chosen at random. What is the probability that the cube is the following: a. black b. red or white c. not blue d. neither red nor white e. Are the events described in parts (b) and (d) complements? Why or why not?

In addition to behind-the-wheel tests, states require written tests before issuing drivers licenses. The failure rate for the written driving test in Florida is about \(60 \%\). (Source: tampabay.com) Suppose three drivers" license test-takers in Florida are randomly selected. Find the probability of the following: a. all three fail the test b. none fail the test c. only one fails the test

A famous study by Amos Tversky and Nobel laureate Daniel Kahneman asked people to consider two hospitals. Hospital \(\mathrm{A}\) is small and has 15 babies born per day. Hospital B has 45 babies born each day. Over one year, cach hospital recorded the number of days that it had more than 609 girls bom. Assuming that \(50 \%\) of all babies are girls, which hospital had the most such days? Or do you think both will have about the same number of days with more than \(60 \%\) girls born? Answer, and explain. (Source: Tversky, Preference, belief, and similarity: Selected Writings, ed. [Cambridge, MA: MIT Press], 205)

A certain professional basketball player typically makes \(80 \%\) of his basket attempts, which is considered to be good. Suppose you go to several games at which this player plays. Sometimes the player attempts only a few baskets, say, \(10 .\) Other times, he attempts about \(60 .\) On which of those nights is the player most likely to have a "bad" night, in which he makes much fewer than \(80 \%\) of his baskets?