Dependent Events

Show your work for each problem.

1.
Which scenario describes dependent events?
1. a)Flipping a coin 3 times
2. b)Drawing 2 cards without replacement
3. c)Rolling a die twice
4. d)Spinning a spinner twice
Answer: Drawing 2 cards without replacement
Drawing cards without replacement is dependent because the first draw removes a card, changing the deck for the second draw. Rolling dice, flipping coins, and spinning spinners are independent.
2.
A bag has 5 red and 3 blue marbles. You draw one marble. What is P(red)?
1. a)
2. b)
3. c)
4. d)
Answer:
There are 5 red marbles and total marbles. So .
3.
A deck has 52 cards. After drawing 1 card, how many cards remain?
Answer: 51
When drawing without replacement, we remove 1 card from 52, leaving cards.
4.
A deck has 13 hearts. After drawing 1 heart, how many hearts remain?
Answer: 12
When we draw 1 heart, only hearts remain in the deck.
5.
From a deck, you draw a heart first. What is P(second card is a heart)?
1. a)
2. b)
3. c)
4. d)
Answer:
After drawing 1 heart: 12 hearts remain out of 51 total cards. So .
6.
A bag has 4 red and 6 blue marbles. Find P(red then blue) without replacement.
Answer: 4/15
1. What is P(red first)? Enter as a fraction. 4/10
2. After removing 1 red, how many marbles remain? 9
3. What is P(blue | red)? Enter as a fraction. 6/9
4. Multiply: 24/90
5. Simplify 4/15
7.
What is ? Simplify completely. Enter as a fraction.
Answer: 1/17
(divide both by 156).
8.
You draw a spade first (not replaced). What is P(second card is a spade)?
1. a)
2. b)
3. c)
4. d)
Answer:
Originally 13 spades in 52 cards. After drawing 1 spade: 12 spades remain in 51 cards. .
9.
A jar has 3 red and 5 green candies. Find P(both green) without replacement.
Answer: 5/14
1. What is the total number of candies? 8
2. What is P(first green)? 5/8
3. After taking 1 green, how many green remain? 4
4. After taking 1 green, how many candies remain total? 7
5. What is P(second green | first green)? 4/7
6. Calculate 20/56
7. Simplify 5/14
10.
A box has 10 items: 8 good, 2 defective. You test 2 items. What is P(both good)? Enter as a fraction.
Answer: 28/45
, . So .
11.
You draw 2 cards. The first is a diamond. What is P(second is a heart)?
1. a)
2. b)
3. c)
4. d)
Answer:
Drawing a diamond doesn't remove any hearts, so there are still 13 hearts. But total cards are now 51. .
12.
A club has 5 seniors and 7 juniors. Two officers are chosen randomly. Find P(both seniors).
Answer: 5/33
1. What is the total number of members? 12
2. What is P(first officer is senior)? 5/12
3. After selecting 1 senior, how many seniors remain? 4
4. After selecting 1 person, how many members remain? 11
5. What is P(second senior | first senior)? 4/11
6. Calculate 20/132
7. Simplify 5/33
13.
From a deck, you draw 2 aces in a row. What is the probability? Enter as a fraction.
Answer: 1/221
, . So .
14.
A bag has 4 red, 3 blue, 2 green balls. Find P(red, then blue, then green) without replacement.
Answer: 1/21
1. What is the total number of balls? 9
2. What is P(red first)? 4/9
3. After removing red, how many balls remain? 8
4. What is P(blue second | red first)? 3/8
5. After removing red and blue, how many balls remain? 7
6. What is P(green third | red, blue)? 2/7
7. Calculate 24/504
8. Simplify 1/21
15.
10 cards dealt from a deck included 2 aces. What is P(next card is an ace)?
1. a)
2. b)
3. c)
4. d)
Answer:
Originally 4 aces. After 10 cards with 2 aces dealt: aces remain in cards. .
16.
Which formula is correct for dependent events?
1. a)
2. b)
3. c)
4. d)
Answer:
For dependent events, we use where is the conditional probability of B given A.