Ex 7.2, 6

Transcript

Ex 7.2, 6 (i) Suppose you roll a 6-sided die 12 times and get a ‘3’ three times. (i) What is the experimental probability of rolling a ‘3’? Given Total rolls = 12 Number of times '3' was rolled = 3 Now, P(rolling a 3) = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 “3” 𝑤𝑎𝑠 𝑟𝑜𝑙𝑙𝑒𝑑)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑜𝑙𝑙𝑠) = 3/12 = 𝟏/𝟒 Ex 7.2, 6 (ii) Suppose you roll a 6-sided die 12 times and get a ‘3’ three times. (ii) What is the theoretical probability of rolling a ‘3’? A die has six sides, and only one of them is a ‘3’ Thus, P(rolling a 3) = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑤𝑖𝑡ℎ “3”)/(𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠) = 𝟏/𝟔 Ex 7.2, 6 (iii) Suppose you roll a 6-sided die 12 times and get a ‘3’ three times. (iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times? They are different because 12 is a very small number of trials, and short-term randomness allows for streaks (like rolling a '3' more often than expected). However, because of the Law of Large Numbers, if we roll the die 60, 600, or 6000 times, we expect the experimental probability to gradually smooth out and get closer and closer to the theoretical probability of 𝟏/𝟔. The more trials you run, the closer your real-world data gets to the perfect mathematical theory.