The objectives of this problem set is to work with the conceptual mechanics of Bayesian data analysis. The target of inference in Bayesian inference is a posterior probability distribution. Posterior probabilities state the relative numbers of ways each conjectured cause of the data could have produced the data. These relative numbers indicate plausibilities of the different conjectures. These plausibilities are updated in light of observations through Bayesian updating.
Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Make sure to include plots if the question requests them.
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2-1. Suppose you have a deck with only three cards. Each card has two sides, and each side is either black or white. One card has two black sides. The second card has one black and one white side. The third card has two white sides. Now suppose all three cards are placed in a bag and shuffled. Someone reaches into the bag and pulls out a card and places it flat on a table. A black side is shown facing up, but you don’t know the color of the side facing down. Show that the probability that the other side is also black is 2/3. Use the counting method (Section 2 of the chapter) to approach this problem. This means counting up the ways that each card could produce the observed data (a black side facing up on the table).
# Card 1: B/W, Card 2: B/B, Card 3 - W/W
# The probability of getting one black side facing up is: Card 1 = 1, Card 2 = 2, Card 3 = 0 -> Total = 1 + 2 = 3
# The probability of getting 2 black sides = 2 black sides probability/total probability = 2/32-2. Now suppose there are four cards: B/B, B/W, W/W, and another B/B. Again suppose a card is drawn from the bag and a black side appears face up. Again calculate the probability that the other side is black.
# Card 1: B/W, Card 2: B/B, Card 3 - W/W, Card 4: B/B
# The probability of getting one black side facing up is: Card 1 = 1, Card 2 = 2, Card 3 = 0, Card 4 = 2 -> Total = 1 + 2 + 2 = 5
# The probability of getting 2 black sides = 2 black sides probability/total probability = 4/52-3. Imagine that black ink is heavy, and so cards with black sides are heavier than cards with white sides. As a result, it’s less likely that a card with black sides is pulled from the bag. So again assume there are three cards: B/B, B/W, and W/W. After experimenting a number of times, you conclude that for every way to pull the B/B card from the bag, there are 2 ways to pull the B/W card and 3 ways to pull the W/W card. Again suppose that a card is pulled and a black side appears face up. Show that the probability the other side is black is now 0.5. Use the counting method, as before.
# Card 1: B/W, Card 2: B/B, Card 3 - W/W
# The probability of getting one black side facing up is: Card 1 = 1*2, Card 2 = 2*1, Card 3 = 0*3 -> Total = 2 + 2 = 4
# The probability of getting 2 black sides = 2 black sides probability/total probability = 2/4 = 0.52-4. Assume again the original card problem, with a single card showing a black side face up. Before looking at the other side, we draw another card from the bag and lay it face up on the table. The face that is shown on the new card is white. Show that the probability that the first card, the one showing a black side, has black on its other side is now 0.75. Use the counting method, if you can. Hint: Treat this like the sequence of globe tosses, counting all the ways to see each observation, for each possible first card.
# Card 1: B/W, Card 2: B/B, Card 3 - W/W
# If the first draw is Card 2, the probability of getting the black side facing up is 2 and the probability of getting the white side in the second draw is 3 (1 way from Card 1 and 2 ways from Card 3)
# If the first draw is Card 1, the probability of getting the black side is 1 and the probability of getting the white side in the second draw is 2 (2 ways from Card 3)
# Total probability = 2*3 + 1*2 = 8
# Getting Card 2 in the first draw and the white side of Card 1 in the second draw is the best outcome which has the probability of 2*3/8 = 6/8 = 0.752-5. Suppose there are two species of panda bear. Both are equally common in the wild and live in the same places. They look exactly alike and eat the same food, and there is yet no genetic assay capable of telling them apart. They differ however in their family sizes. Species A gives birth to twins 10% of the time, otherwise birthing a single infant. Species B births twins 20% of the time, otherwise birthing singleton infants. Assume these numbers are known with certainty, from many years of field research. Now suppose you are managing a captive panda breeding program. You have a new female panda of unknown species, and she has just given birth to twins. What is the probability that her next birth will also be twins?
# First birth
# Probability of being A is 0.5 and of being B is 0.5
# Probability of being A and giving birth to twins is 0.5*0.1=0.05
# Probability of being B and giving birth to twins is 0.5*0.2=0.1
# Total chance = 0.05+0.1=0.15
# Second birth
# Probability of being A and giving second birth to twins is 0.05/0.15*0.1=1/30
# Probability of being B and giving second birth to twins is 0.1/0.15*0.2=4/30=2/15
# Total chance = 1/30+2/15=1/6=0.1667