Randomness, Probability, and Simulation

In probability and statistics, a random phenomenon is 

• something that is completely unexpected or surprising
• something that has a limited set of outcomes, but when each outcome occurs is completely unpredictable.
• something whose outcome defies description.
• something that is unpredictable from one occurrence to the next, but over the course of many occurrences follows a predictable pattern 

The probability that a randomly-selected person is left-handed is about 1/10.  Which of the following statements must be true?

• If you randomly selected ten people, exactly one will be left-handed.
• In any randomly-selected group of 10000 people, exactly 1000 of them will be left-handed.
• As you randomly choose more and more people, the proportion of left-handers may start out quite far from 1/10, but it will get close to 1/10 as the number of people you select increases.
• all three are true.

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You are playing a board game with some friends that involves rolling two six-sided dice.  For eight consecutive rolls, the sum on the dice is 6.  Which of the following statements is true?

• Each time you roll another 6, the probability of getting yet another 6 on the next roll goes down.
• Each time you roll another 6, the probability of getting yet another 6 on the next roll goes up.
• You should find another set of dice:  eight consecutive 6’s is impossible with fair dice.
• The probability of rolling a 6 on the ninth roll is the same as it was on the first roll.

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Your family moves into a new home with your beloved greyhound “Max.”  You discover that the family next door also has a greyhound named “Max.”  Is this so surprising that you should call the newspapers?

• Yes – the probability of this happening is infinitesimally small.
• Yes – though it would not be surprising if the dogs had been a more popular breed, such as Labrador retrievers.
• No – but only because the dogs are greyhounds: many greyhounds have been “rescued” from the dog-racing industry, and “Max” is a popular name for racing dogs.
• No – while this particular event might seem remarkable, the probability that it occurs somewhere in the country at some point is quite high.

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As the number of tosses of a fair coin goes up from 10, to 100, to 1,000 and to 10,000, what happens to the probability of getting between 40% and 60% heads? What happens to the probability of getting exactly 50% heads?

• Both of those probabilities increase.
• Both of those probabilities decrease.
• The first probability increases, but the second one decreases.
• The first probability decreases, but the second one increases.

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A poker player is dealt poor hands for several hours. He decides to bet heavily on the last hand of the evening on the grounds that after many bad hands he is due for a winner.

• He's right, because the winnings have to average out.
• He's wrong, because successive deals are independent of each other.
• He's right, because successive deals are independent of each other.
• He's wrong, because he’s clearly on a “cold streak.”

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A basketball player makes 47% of her shots from the field during the season. To simulate whether a shot hits or misses you would assign random digits as follows:

• One digit simulates one shot; 4 and 7 are a hit, other digits are a miss.
• One digit simulates one shot; odd digits are a hit and even digits are a miss.
• Two digits simulate one shot; 00 to 47 are a hit and 48 to 99 are a miss.
• Two digits simulate one shot; 00 to 46 are a hit and 47 to 99 are a miss.

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Use the correct choice from the previous question and these random digits to simulate 10 shots:  82734 71490 20467 47511 81676 55300 94383 14893 How many of these 10 shots are hits?

• 2
• 3
• 4
• 5

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You want to estimate the probability that the player makes 5 or more of 10 shots. You simulate 10 shots 25 times and get the following numbers of hits: 5 7 5 4 1 5 3 4 3 4 5 3 4 4 6 3 4 1 7 4 5 5 6 5 7 What is your estimate of the probability?

• 5/25, or 0.20
• 11/25, or 0.44
• 12/25, or 0.48
• 16/25, or 0.64

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Place each word in the proper position for a classical probability?

• Frequency/Event
• Event/Sample Space
• Event/Frequency
• Sample Space/Event

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