A lady wishes to color her fingernails on one hand using at most two of the colors red, yellow, and blue. How many ways can she do this?
How many ways can six indistinguishable letters be put in three mail boxes? Hint: One representation of this is given by a sequence [math]|[/math]LL[math]|[/math]L[math]|[/math]LLL[math]|[/math] where the [math]|[/math]'s represent the partitions for the boxes and the L's the letters. Any possible way can be so described. Note that we need two bars at the ends and the remaining two bars and the six L's can be put in any order.
A travel bureau estimates that when 20 tourists go to a resort with ten hotels they distribute themselves as if the bureau were putting 20 indistinguishable objects into ten distinguishable boxes. Assuming this model is correct, find the probability that no hotel is left vacant when the first group of 20 tourists arrives.
An elevator takes on six passengers and stops at ten floors. We can assign two different equiprobable measures for the ways that the passengers are discharged: (a) we consider the passengers to be distinguishable or (b) we consider them to be indistinguishable (see Exercise for this case). For each case, calculate the probability that all the passengers get off at different floors.
You are playing heads or tails with Prosser but you suspect that his coin is unfair. Von Neumann suggested that you proceed as follows: Toss Prosser's coin twice. If the outcome is HT call the result win. if it is TH call the result lose. If it is TT or HH ignore the outcome and toss Prosser's coin twice again. Keep going until you get either an HT or a TH and call the result win or lose in a single play. Repeat this procedure for each play. Assume that Prosser's coin turns up heads with probability [math]p[/math].
- Find the probability of HT, TH, HH, TT with two tosses of Prosser's coin.
- Using part (a), show that the probability of a win on any one play is 1/2, no matter what [math]p[/math] is.
John claims that he has extrasensory powers and can tell which of two symbols is on a card turned face down (see Example). To test his ability he is asked to do this for a sequence of trials. Let the null hypothesis be that he is just guessing, so that the probability is 1/2 of his getting it right each time, and let the alternative hypothesis be that he can name the symbol correctly more than half the time. Devise a test with the property that the probability of a type 1 error is less than .05 and the probability of a type 2 error is less than .05 if John can name the symbol correctly 75 percent of the time.
In Example assume the alternative hypothesis is that [math]p = .8[/math] and that it is desired to have the probability of each type of error less than .01. Use the program PowerCurve to determine values of [math]n[/math] and [math]m[/math] that will achieve this. Choose [math]n[/math] as small as possible.
A drug is assumed to be effective with an unknown probability [math]p[/math]. To estimate [math]p[/math] the drug is given to [math]n[/math] patients. It is found to be effective for [math]m[/math] patients. The method of maximum likelihood for estimating [math]p[/math] states that we should choose the value for [math]p[/math] that gives the highest probability of getting what we got on the experiment. Assuming that the experiment can be considered as a Bernoulli trials process with probability [math]p[/math] for success, show that the maximum likelihood estimate for [math]p[/math] is the proportion [math]m/n[/math] of successes.
Recall that in the World Series the first team to win four games wins the series. The series can go at most seven games. Assume that the Red Sox and the Mets are playing the series. Assume that the Mets win each game with probability [math]p[/math]. Fermat observed that even though the series might not go seven games, the probability that the Mets win the series is the same as the probability that they win four or more game in a series that was forced to go seven games no matter who wins the individual games.
- Using the program PowerCurve of Example find the probability that the Mets win the series for the cases [math]p = .5[/math], [math]p = .6[/math], [math]p =.7[/math].
- Assume that the Mets have probability .6 of winning each game. Use the program PowerCurve to find a value of [math]n[/math] so that, if the series goes to the first team to win more than half the games, the Mets will have a 95 percent chance of winning the series. Choose [math]n[/math] as small as possible.