Math 10B: Homework 14 (probability) Due: March 13th 1. Suppose I repeatedly roll a fair die, and stop once I roll some number twicein-a-row. What's the expected number of rolls needed? (Hint: For each roll except the first, what would you consider a \"success\"?) 2. If you're dealt a bridge hand (13 cards) at random from a standard 52-card deck, what's the expected value of the number of different suits in your hand? 3. We roll 2 6-sided dice 100 times and record the outcomes for the sum of the dice in the following table. Value 2 3 4 5 6 7 8 9 10 11 12 Total Observed Frequency 6 10 9 13 13 12 11 10 7 5 4 100 Calculate the expected frequencies, given the null hypothesis H0 that both dice are fair. Compute the 2 statistic for this data. What is the p-value? Do we have enough evidence to reject the null hypothesis? 4. Same problem, but with the data from the following table: 1 Value 2 3 4 5 6 7 8 9 10 11 12 Total Observed Frequency 6 10 9 11 13 17 8 10 5 6 5 100 5. This table records the observed frequencies of joint outcomes for a random variable X taking the 3 values x1 , x2 , x3 and a random variable Y taking the 2 values y1 , y2 . y1 y2 x1 x2 x3 827 159 184 177 239 135 Construct a table showing the expected frequencies under the hull hypothesis H0 that X and Y are independent. Find the 2 test statistic and compute its p-value. Do we have enough evidence to reject the null hypothesis? 6. Same problem, but for a random variable Y that takes on 3 values and for these observed data: y1 y2 y3 x1 x2 x3 16 29 23 17 62 34 77 42 15 2