Question 6

The local casino has a new game that involves tossing two identical coins. The fee for playing the game is $6. If the outcome of the two coins is 2 heads the player receives $18. For the outcome of 1 head and 1 tail. the player receives $8. If the outcome is 2 tails. the player receives $0. The coins are specially made so that the chance of obtaining a tail is 70%. The outcomes of the two coins are independent of each other.

On average, is the player expected to make a profit or a loss in the game? In the answer box, please enter P for profit. L for loss and N if it is neither a profit nor a loss.

John played the game 10 times. Calculate his expected profit or loss in the 10 games to 2 decimal places and enter your answer in the answer box. Please omit the + or – sign as well as the $ sign.

Suppose X is a discrete random variable that takes on five possible values, namely, 1. 3, 5. 7 and U. The value of U is to be determined from the information given later.

The probability that X takes on some of the values are listed below.

a) What is the probability that X takes on the value of U? That is, find P(X = U). Give your answer to 2 decimal places in the answer box. Answer b) If the expected value or mean of X is exactly 6, determine the value of U. Give your answer to 2 decimal places in the answer box. Answer

c) Find the variance of X. Give your answer to 2 decimal places in the answer box. Answer.

d) Determine the mode of the distribution to 2 decimal places, if it is a unique value. If the mode is not unique, enter N in the answer box.

Recall that the distribution function F is defined as F(t) = P(X < t). e) Determine F(5.2). Give your answer to 2 decimal places in the answer box.

Question 12

The table below tabulates the number of students in a particular course according to their sex and preference for apples.

Like apples, Indifferent Dislike apples Total Girls 12 10 36 58 Boys 7 16 19 42 Total 19 26 55 100

a) If a student is selected at random, what is the probability that the student likes apples? Give your answer to 2 decimal places in the answer box.

b) What is the probability that a student selected at random likes apples given that the student is a boy? In the answer box, fill in your answer to 2 decimal places.

c) What is the probability that a student selected at random is a girl or is indifferent to apples? In the answer box. provide your answer to 2 decimal places.

Question 13

In a TV game show, a contestant is presented with four envelopes of which one contains $10,000 and the other three contain $100 each. At the start, the contestant selects one of the envelopes and the host of the show will open one of the unselected envelopes to reveal that it contains $100. The host knows which envelope contains the $10,000 and will not open that envelope. After this, the contestant can switch his or her initial selection to one of the other unopened envelopes or else retain his or her initial selection.

Let us label these two choices as ‘switch strategy’ and ‘retain strategy.
Switch strategy: abandons his or her initial selection and switches to one of the remaining envelopes at random (with equal probability). Retain strategy: sticks with his or her initial selection.

As an illustration, suppose the envelopes are labeled A, B. C and D. Suppose the contestant selects envelope B initially. The game host knows which envelope contains the $10,000, and so opens the envelope, say D, to reveal $100. The contestant can now switch his or her selection to either envelope A or C (switch strategy) or else keep to his or her initial selection of envelope B (retain strategy).

Assume that at the start, the contestant selects one of the envelopes randomly. That is, each envelope has an equal chance of being selected at the beginning.

a) Determine the probability of winning $10,000 with the switch strategy. Give your answer to 3 decimal places in the answer box.

Answer b) Determine the probability of winning $10,000 with the retain strategy. Give your answer to 3 decimal places in the answer box.

Question 14

A data set comprises twelve observations which are listed below:

236. 292. 332. 294. 313. 289. 271. 207. 325. 205. 295. 285.

a) Compute the sample standard deviation. Give your answer to 2 decimal places in the answer box.

Both methods for computing the quartiles or percentiles that were discussed in class (percentile.exc and percentile.inc) are acceptable. Please apply one of the methods consistently for the following two questions.

b) Determine the inter•quartile range of the data set. Give your answer to 2 decimal places in the answer box.

c) Determine the 80th percentile. Give your answer to 2 decimal places in the answer box.