Prob. Rand. Var. Tutorial#1 PDF

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2.1 An experiment consists of rolling n (six-sided) dice and recording the sum of the n rolls. How many outcomes are there to this experiment? Solution: For 𝑛 = 1 die, there are 6 possible outcomes of the sum (1-6). For 𝑛 = 2 dice, there are 11 possible outcomes of the sum (2-12). For 𝑛 = 3 dice, you can obtain a sum from 3-18, which is 16 possible outcomes. The general case for n dice, there are 6𝑛 − 𝑛 + 1 cases. This is equivalent to 5𝑛 + 1. 2.2 (a) An experiment consists of rolling a die and flipping a coin. If the coin flip is heads, the value of the die is multiplied by -1, otherwise it is left as is. What are the possible outcomes of this experiment? (b) Now, suppose we want to repeat the experiment in part (a) n times and record the sum of the results of each experiment. How many outcomes are there in this experiment and what are they? Solution: (a) If the coin is heads, then the outcome could be -1, -2, -3, -4, -5, or -6. If the coin is tails, then the outcome could be 1, 2, 3, 4, 5, 6. These 12 options are the total possible outcomes of the experiment. (b) If you repeat the experiment 2 times (𝑛 = 2), then the sum of the dice could range from -12 to 12 (or −6𝑛 to 6𝑛). For 𝑛 = 3 times, the values of -18 to 18 could be obtained. Thus for 𝑛 times, there are 12𝑛 + 1 outcomes which range from −6𝑛 to 6𝑛 . 2.5 An experiment consists of selecting two integers (𝑛, 𝑘) such that a) How many outcomes are in the sample space? b) What fraction of the outcomes in the sample space satisfy c) What fraction of the outcomes in the sample space satisfy d) What fraction of the outcomes in the sample space satisfy

0 ≤ 𝑛 < 5 and 0 ≤ 𝑘 < 10 . 𝑛>𝑘? 𝑛...