Probability

A Dutch Book is some mix of wagers, which can be shown to result in a sure loss (according to the bettor’s beliefs). Consider this situation (The three-card problem): there are three cards, one is red on both sides (RR), the second is red on one side and white on the other (RW), and the third card is white on both sides (WW). A single card is drawn (eyes closed) and tossed into the air.

a. P (RR card selected) = ? b. P (W-showing) = ? c. P (not – RR|R – showing) = ?

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d. Are the following fair bets or a Dutch Book: i. You pay $1 – to guess a card You win $3 – if correct ii. R-showing You pay $1 – to guess card You win $2 – if correct iii. Win $1– if R-showing and not RR You lose $1– if R-showing and RR

A gambling bet is considered fair if the individual making the bet considers that they will break even in the long run.

Which of the following bets would be considered fair? a. A fair coin is flipped. You pay $1 to guess the outcome and receive $2 in return if your guess is correct. b. You pay $5 to toss two dice. If the total is 7 or 11 you receive $20 in return.

Suppose that the people attending a resort have a long-term chance of 1 in 100 of having skin cancer (from too much sun exposure). The resort maintains a clinic to help detect this disease. Suppose that the screening used at this clinic has a false positive rate of 0.2 (that is 20%) of people without skin cancer will test positive for cancer) and that it has a false negative rate of 0.1 (10% of people with skin cancer will test negative). Suppose that a person has just tested positive for skin cancer.

What is the probability that he actually has the disease?

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