Question

compound events
the probability of two or more simple events.
independent events
when the outcome of one event does not affect the outcome of the other event.
p(a and b) = p(a)·p(b)
dependent events
when the outcome of one event does affect the outcome of the other event.
p(a and b) = p(a)·p(b after a occurs)
independent events

1. a die is rolled 3 times. what is the probability of getting 1’s on each roll?
2. a coin is tossed, then a day of the week is selected. what is the probability of getting tails then a day starting with the letter t?

a bag contains 8 red crayons, 14 purple crayons, 6 yellow crayons, and 4 green crayons. a crayon is selected, replaced, then another is selected. find each probability.

1. p(purple then yellow)
2. p(green then red)
3. p(two purples)
4. p(two yellows)

dependent events
using the same example from above, assume once a crayon is selected, it is not replaced. find each probability.

1. p(yellow then red)
2. p(purple then green)
3. p(two reds)
4. p(two greens)
5. a card is drawn from a standard deck, not replaced, and another is drawn. what is the probability of choosing a heart then a spade?
6. jack had four snicker bars and 8 mars bars. he randomly chose a piece of candy, ate it, then chose another. what is the probability that both candy bars were snickers?

Explanation:

Step1: Identify total crayons

Total crayons = 8 (red) + 14 (purple) + 6 (yellow) + 4 (green)? Wait, wait, original problem: 8 red, 14 purple, 6 yellow, 4 green? Wait, no, the table says "8 red crayons, 14 purple crayons, 6 yellow crayons, and 4 green crayons"? Wait, total is 8 + 14 + 6 + 4 = 32. Correct.

Step2: Probability of first event (purple)

For independent events (replaced), P(purple) = number of purple / total = 14/32.

Step3: Probability of second event (purple)

Since replaced, P(purple) again is 14/32.

Step4: Multiply for independent events

P(two purples) = (14/32) (14/32) = (196)/(1024) = 49/256. Wait, wait, 1414=196, 32*32=1024. Simplify: divide numerator and denominator by 4: 49/256. Wait, but let's check the steps again.

Wait, the problem 15 is P(two purples) with replacement (since it's under independent events, which are replaced). So:

Total crayons: 8 + 14 + 6 + 4 = 32. Correct.

Number of purple: 14.

So P(first purple) = 14/32.

Since replaced, P(second purple) = 14/32.

So multiply: (14/32) (14/32) = (1414)/(32*32) = 196/1024 = 49/256 (divided numerator and denominator by 4: 196 ÷4=49, 1024 ÷4=256).

Answer:

$\frac{49}{256}$