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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) In Example, a mathematical model is discussed which claims to give a distribution of identical balls into boxes in such a way that all distinguishable arrangements are equally probable (Bose-Einstein statistics). Prove this by showing that the probability of a distribution of N balls into n boxes (according to this model) with balls in the first box, in the second, 路路路 , in the , is for any set of numbers Ni such that.
b) Show that the model in (a) leads to Maxwell-Boltzmann statistics if the drawn card is replaced (but no extra card added) and to Fermi-Dirac statistics if the drawn card is not replaced. Hint: Calculate in each case the number of possible arrangements of the balls in the boxes. First do the problem of particles in boxes as in the example, and then do N particles in n boxes ( ) to get the results in Problem .
A single card is drawn at random from a shuffled deck. What is the probability that it is red? That it is the ace of hearts? That it is either a three or a five? That it is either an ace or red or both?
Two people are taking turns tossing a pair of coins; the first person to toss two alike wins. What are the probabilities of winning for the first player and for the second player? Hint: Although there are an infinite number of possibilities here (win on first turn, second turn, third turn, etc.), the sum of the probabilities is a geometric serieswhich can be summed; see Chapter 1 if necessary.
Two cards are drawn at random from a shuffled deck.
What is the probability that at least one is a heart?
(b) If you know that at least one is a heart, what is the probability that both are
hearts?
A weighted coin with probability p of coming down heads is tossed three times; x = number of heads minus number of tails.
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