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Chapter 15: Q3P (page 755)
A ball is thrown straight up and falls straight back down. Find the probability density function so that is the probability of finding it between height and . Hint: Look at Example .
The probability density function is .
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(a) Find the probability density function for the position x of a particle which is executing simple harmonic motion on along the x axis. (See Chapter , Section , for a discussion of simple harmonic motion.) Hint: The value of x at time t is . Find the velocity ; then the probability of finding the particle in a given is proportional to the time it spends there which is inversely proportional to its speed there. Don’t forget that the total probability of finding the particle somewhere must be .
(b) Sketch the probability density function found in part (a) and also the cumulative distribution function [see equation ].
(c) Find the average and the standard deviation of x in part (a).
A trick deck of cards is printed with the hearts and diamonds black, and the spades and clubs red. A card is chosen at random from this deck (after it is shuffled). Find the probability that it is either a red card or the queen of hearts. That it is either a red face card or a club. That it is either a red ace or a diamond.
Two cards are drawn from a shuffled deck. What is the probability that both are red? If at least one is red, what is the probability that both are red? If at least one is a red ace, what is the probability that both are red? If exactly one is a red ace, what is the probability that both are red?
(a) Set up a sample space for the 5 black and 10 white balls in a box discussed above assuming the first ball is not replaced. Suggestions: Number the balls, say 1 to 5 for black and 6 to 15 for white. Then the sample points form an array something like (2.4), but the point 3,3 for example is not allowed. (Why?
What other points are not allowed?) You might find it helpful to write the
numbers for black balls and the numbers for white balls in different colors.
(b) Let A be the event “first ball is white” and B be the event “second ball is
black.” Circle the region of your sample space containing points favorable to
A and mark this region A. Similarly, circle and mark region B. Count the
number of sample points in A and in B; these are and . The region
AB is the region inside both A and B; the number of points in this region is
. Use the numbers you have found to verify (3.2) and (3.1). Also find
and and verify (3.3) numerically.
(c) Use Figure 3.1 and the ideas of part (b) to prove (3.3) in general.
Given the measurements.
Find the mean value and the probable error of and .What do you think about this solution?
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