1. Click through probability
Your webpage is one of several search results matching a particular query. Your page is the second result down.
The user will read down the list in order and at each link decide whether or not to click on that result with probability 0.5. Once they have left the result page (by clicking on a result), they will return to the result page with probability 0.4 and keep reading down the list starting from below the result they last visited.

• Draw a probability tree showing the user's behavior.
• What is the probability they will click on the first result?
• What is the probability they will click on the second (your) result, given that they have clicked on the first result?
• What is the total probability they will click on the second (your) result?

2. Overlapping specializations

The probability that a student selects the HCI specialization is 25%. The probability that a student selects the social computing specialization is 5%. The probability that a student selects the social computing specialization given that they have selected HCI is 10%. What is the probability that someone is specializing in HCI, given that they are specializing in social computing? Explain your reasoning.

3. Biased coins

For this problem you will attempt to bias a coin. You may alter the coin (or checker or other small disk-like object) as you wish, for example by filing it or sticking putty or gum on it or by selecting something disk-like but asymmetrical. You'll try two experiments:

• spinning the coin rapidly and recording which side is up when it falls over
• hold the coin with the coin's face parallel to the ground, toss it in the air, making sure it flips a few times, and catch it mid-air in the palm of your hand.

Do each of these 20 times. If you wish you can team up (share the coin) with up to 3 other students and get more observations in. How many times did "heads" turn up for the spin? For the toss? Using the pbinom() function in R calculate separately for the spins and tosses what the probability would be of observing by chance that number of "heads" or fewer (if "tails" came up more often, otherwise substitute "tails" for "heads") with an unbiased coin.

4. Hiring bias

There was a staff position opening, with 10 equally qualified applicants. 4 of the applicants were women, and 6 were men. However, all 3 of the applicants that were interviewed were women. If the process was gender-unbiased, what would be the probability that 3 women were interviewed? Would you conclude that the hiring process was biased? Explain.