Consider two games at a village fair, A and B. Let the gain in dollars in one play of Game A be denoted by random variable X, and let the gain in dollars in one play of Game B be denoted by random variable Y. (A negative gain means that you receive less money back from the game than you paid to take part.) Probability distributions for X and Y are shown below. X -5 -2 1 4 10 Probability 0.33 0.27 0.2 0.13 0.07 Y -5 -2 1 4 25 Probability 0.37 0.32 0.16 0.10 0.05 The results of each game are independent of each other. (a) In one play of Game A, what is the probability of a positive gain? How about the probability of a positive gain in Game B? (b) The expected value and the standard deviation of money gained in one play of Game A are $-0.77 and $4.24, respectively. Calculate the expected value and the standard deviation of the money gained in one play of Game B. (c) Suppose you are contemplating making a lot of plays in Game A or Game B. To see which of these is smarter financially, you ask a friend, and he says that Game A should be your choice because your gain is more likely to be positive. Is this correct logic? Explain why or why not.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Consider two games at a village fair, A and B. Let the gain in dollars in one play of Game A be denoted by random variable X, and let the gain in dollars in one play of Game B be denoted by random variable Y. (A negative gain
Probability distributions for X and Y are shown below.
| X | -5 | -2 | 1 | 4 | 10 |
| Probability | 0.33 | 0.27 | 0.2 | 0.13 | 0.07 |
| Y | -5 | -2 | 1 | 4 | 25 |
| Probability | 0.37 | 0.32 | 0.16 | 0.10 | 0.05 |
The results of each game are independent of each other.
(a) In one play of Game A, what is the probability of a positive gain? How about the probability of a positive gain in Game B?
(b) The
(c) Suppose you are contemplating making a lot of plays in Game A or Game B. To see which of these is smarter financially, you ask a friend, and he says that Game A should be your choice because your gain is more likely to be positive. Is this correct logic? Explain why or why not.
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