Probability of Two Draws: Replacement, Conditional Events, and Frequency

All flashcards

What is theoretical probability?

A probability calculated from a model of possible outcomes and their likelihoods, rather than from a particular set of observed trials.

What is empirical probability?

An estimate based on observed results: the number of trials in which an event occurs divided by the total number of trials.

How does replacement affect two successive picks?

With replacement, the collection is restored before the second pick, so the second-pick probabilities do not change based on the first pick. Without replacement, the first candy remains out, leaving 39 candies and making the picks dependent.

What does $P(G_2\mid R_1)$ mean?

It is the probability of green on the second pick given that red occurred on the first pick. The condition restricts attention to outcomes where the first pick was red.

What is long-term relative frequency?

It is the pattern in which an event's observed relative frequency tends to settle nearer its theoretical probability as the number of comparable trials grows. It does not guarantee an exact match in any particular set of trials.

Quiz questions

1. A collection has 40 candies, including $r$ red candies. What is the probability of drawing two reds without replacement?Answer: $(r/40)((r-1)/39)$
2. A collection has 40 candies, of which $y$ are yellow. What is the probability of drawing no yellows in two picks with replacement?Answer: $((40-y)/40)^2$
3. A collection has six color counts $c_1,c_2,\ldots,c_6$. Which expression gives the probability of doubles with replacement?Answer: $\sum_{i=1}^{6}(c_i/40)^2$
4. A collection has 40 candies, including $g$ green candies. Without replacement, what is $P(G_2\mid R_1)$?Answer: $g/39$
5. In two ordered picks, what event does $P(G_1\text{ AND }R_2)$ describe?Answer: Green on the first pick and red on the second pick.