C955 Applied Probability and Statistics - Set 3 - Part 2

Correct Answer: n times p times (1 minus p)

The variance of a binomial distribution is n times p times (1 minus p), which is largest when p equals 0.5.

Correct Answer: 50

The binomial mean n times p equals 100 times 0.5, which is 50.

Correct Answer: As sample size grows, the sample mean approaches the true mean

The law of large numbers states that as the number of trials increases, the observed average converges toward the theoretical expected value.

Correct Answer: Over many bets the average result approaches the built-in house edge

Although individual outcomes vary, over the enormous number of bets played the casino's average result reliably reflects its statistical edge.

Correct Answer: Believing past independent outcomes affect future ones

The gambler's fallacy is the mistaken belief that a run of one outcome makes the opposite more likely next, ignoring that independent trials have no memory.

Correct Answer: 0.5

Coin flips are independent, so previous outcomes do not affect the next flip. The probability of heads remains 0.5 regardless of the streak.

Correct Answer: Zero net gain in the long run

A fair game has an expected value of zero, meaning that over many plays the average net gain or loss is zero.

Correct Answer: -$1

Expected winnings are 1000 times 0.001, which is $1. Subtracting the $2 cost gives an expected net value of -$1 per ticket.

Correct Answer: The player loses money on average over time

A negative expected value means that, averaged over many plays, the player loses money, which is typical of commercial gambling and lottery games.

Correct Answer: Visualizing sequential events and computing combined probabilities

A tree diagram maps sequential events with branch probabilities, and multiplying along a path gives the probability of that sequence of outcomes.

Correct Answer: A reversed conditional probability using known information

Bayes' theorem updates the probability of a cause given observed evidence, reversing a known conditional probability using prior information.

Correct Answer: When the disease is rare, false positives can outnumber true positives

With a low base rate, even a small false positive rate applied to the large healthy population can produce more false positives than true positives, a key Bayesian insight.

Correct Answer: The probability of an event before new evidence is considered

The prior is the initial probability assigned before observing new data, which Bayes' theorem then updates into a posterior probability.

Correct Answer: Its overall frequency in the population

The base rate is how common an outcome is in the population overall, and neglecting it leads to the base rate fallacy in interpreting test results.

Correct Answer: Ignoring overall prevalence when interpreting specific evidence

The base rate fallacy occurs when people focus on specific test accuracy while ignoring how rare the condition is, leading to overestimated probabilities.

Correct Answer: The probability that two events both occur

A joint probability is the likelihood that two events occur together, written P(A and B).

Correct Answer: The probability of a single event regardless of others

A marginal probability is the probability of one event on its own, obtained by summing joint probabilities across the other variable's outcomes.

Correct Answer: 2/52

There are two red kings (hearts and diamonds) in the deck, so the probability is 2 out of 52, which simplifies to 1/26.

Correct Answer: 28/52

There are 26 red cards and 4 kings, but 2 kings are already red. Using the addition rule, 26 plus 4 minus 2 equals 28, giving 28/52.

Correct Answer: Each integer has probability 1/6

A discrete uniform distribution assigns equal probability to each of the six outcomes, so each has probability 1/6 and they sum to 1.