There are 12 cartons of eggs.
There are 12 cartons of eggs. We randomly pick one egg from each carton and put it in a large bowl of water to test the freshness. Fresh eggs sink to the bottom of the bowl and rotten ones float. How many possible freshness sequences are possible?
(b) Randomly pick 4 eggs from a carton of 18 eggs, 6 of which cracked.
I) Suppose we label the eggs from 1 to 18, how many different ways to pick 4 eggs?
ii) Suppose we only care whether a selected egg is cracked, what is the probability that two cracked eggs will be selected?
Roulette is a game named after a French diminutive for little wheel.
Roulette is a game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even. The 18 red, 18 block, and 2 green pockets of the roulette wheel are numbered from 1 to 36, 0, and 00. The payout is 1 to 1 if you bet on red or block, 1 to 1 if you bet on even or odd numbers, 35 to 1 if you be on one number among 38 available numbers.
(a) Suppose the outcome of interest is the colour of the pocket in which the ball falls. List all possible outcomes for three games. Are the elements in the sample space equally likely?
(b) Find the probability distribution of the number of games that the ball falls into a red pocket for three games.
(c) Let X be the number of games that the ball falls into a even-numbered pocket in the next 20 games. Verify that X is a binomial random variable. Note that 0 and 00 don’t count as an even number in this game.
(d) Determine the mean and variance of X.
(e) What is the probability that there will be no more than 17 games that the ball falls into a red pocket in the following 20 games.
Data show that, on average, 6.9 patients arrive at the emergency room
Data show that, on average, 6.9 patients arrive at the emergency room of a large hospital each hour. In that hospital, Doctors at the emergency room work four hours as a shift.
(a) Determine the mean and variance of the number of patients that a doctor has during a shift.
(b) Find the probability that
1) exactly five patients will arrive at the emergency room during a one-hour period.
2) within 27 to 29 patients will arrive at the emergency room during a doctor’s shift.
3) at most two patients will arrive the emergency room during the last hour in a doctor’s shift. Is it unusual?
Based on golf club sales data from a number of equipment
Based on golf club sales data from a number of equipment manufac- turers, about 30% of Canadians play golf left handed. Would it be unusual to find less than two golfers are right handed in a random sample of five golfers.
