set.seed(12345)
coin <- c("Heads", "Tails")
tosses <- sample(coin, 10, replace = TRUE)
data.frame(tosses) |>
group_by(tosses) |>
summarise(n = n())
# A tibble: 2 × 2
tosses n
<chr> <int>
1 Heads 3
2 Tails 7
xkcd comic showing two people discussing what it means to have a 50-50 chance
Let Ω be the outcome space, and let P(A) denote the probability of the event A. Then we have:
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The Linda problem is from a very famous experiment conducted by Daniel Kahneman and Amos Tversky in 1983 (The version below is from the book Thinking, Fast and Slow by Kahneman, page 156):
Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.
Which alternative is more probable?
Linda is a bank teller.
Linda is a bank teller and is active in the feminist movement.
Kahneman, Daniel. Thinking, Fast and Slow (p. 158). Farrar, Straus and Giroux.
set.seed(12345)
tosses <- sample(coin, 500, replace = TRUE)
data.frame(tosses) |>
group_by(tosses) |>
summarise(n = n())
# A tibble: 2 × 2
tosses n
<chr> <int>
1 Heads 251
2 Tails 249
We see that as the number of tosses increases, the split of heads and tails begins to look closer to 50-50.
Here is a plot of the proportion of tosses that land heads when we toss a coin n times, where n varies from 1 to 1000.
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Suppose Ali and Bettina are playing a game, in which Ali tosses a fair coin n times, and Bettina wins one dollar from Ali if the proportion of heads is less than 0.4. Ali lets Bettina decide if n is 10 or 100.
Which n should Bettina choose?
Part 1: Suppose we roll a die 4 times. The chance that we see six (the face with six spots) at least once is given by 16+16+16+16=46=23
True or false?
Part 2: Suppose we roll a pair of dice 24 times. The chance that we see a pair of sixes at least once is given by 24×136=2436=23
True or false?
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Consider the Venn diagram below, which has 20 possible outcomes in Ω, depicted by the purple dots. Suppose the dots represent equally likely outcomes. What is the probability of A or B or C? That is, what is P(A∪B∪C)?
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