Probability: How to Solve

Three coins are tossed up in the air, one at a time. What is the probability that two of them will land heads up and one will land tails up?

How do you solve this?

Probability space. Do 2 × 2 × 2 = 8 to find the number of combinations (outcomes times number of tosses). Then write out the combos to find how many: e.g. HHH HTT

Using a six-sided die, Carlin has rolled a six on each of 4 successive tosses. What is the probability of Carlin rolling a six on the next toss

Think the die has no memory/the rolls are independent. So it is 1/6

A regular deck of cards has 52 cards. Assuming that you do not replace the card you had drawn before the next draw, what is the probability of drawing three aces (of which there are 4 in a deck) in a row?

Use multiplication rules and subtract out cards because there is no replacement. So it is 4/52 × 3/51 × 2/50

An MP3 player is set to play songs at random from the 15 songs it contains in memory. Any song can be played at any time, even if it is repeated. There are 5 songs by Band A, 3 songs by Band B, 2 by Band C, and 5 by Band D. If the player has just played two songs in a row by Band D, what is the probability that the next song will also be by Band D?

Each play is independent so it is just 5/15 (the total number of songs)

An MP3 player is set to play songs at random from the fifteen songs it contains in memory. Any song can be played at any time, even if it is repeated. There are 5 songs by Band A, 3 songs by Band B, 2 by Band C, and 5 by Band D. What is the probability that the next two songs will both be by Band B?

As you have 2 songs, you need to do multiplication rule. 3/15 × 3/15.

Alan walks or bikes to school. Probability that he bikes to school on a given day is .75. If he bikes the probability he is late is .05. If he walks the probability of lateness is .1.

So, probability he bikes = .75

Probability he walks = .25

Probability he is on time biking = .95

Probability he is on time walking = .90

What is the probability he is late? .25 * .1 = walks and was late. .75 x .05 = cycled and was late. Sum of these two probabilities is the probability he is late overall.

Suppose that A and B are two independent events. What is P(A or B)?

Independent means they are not disjoint, so you’ll do P(A) + P(B) - P(A and B)

In NOLA, 72% have a phone, 38% a pager, and 29% both. What percent of people in NOLA have both a phone and pager?

You can just add 72 and 38 and subtract 29. No need to do extra arithmetic. Note these events are independent so you need to minus out the 29%.

Consider two events: E and F. P(E)=P(F)=.7
Suppose we know that P(F|E)=.9. What is the probability that at least one of them occurs? P(E or F)?

This when you use non-independent multiplication rule. You do .7 + .7 - (.7 x .9). The .7 x .9 is P(A) * P(B|A).