TACHS: Two draws without replacement: update the bag
Find the probability of two draws in a row when the first item is not put back, including one of each, at least one, and draws with replacement.
How do you find the probability of drawing two things in a row when the first one is not put back?
Short answer
Multiply the probability of the first draw by the probability of the second draw from the changed bag. Without replacement, the total drops by 1, and so does the count of the color drawn first.
Know first: Probability as favorable ÷ total, Multiplying and simplifying fractions
Step 1 of 6
The bag changes after the first draw
The probability of an event is favorable outcomes ÷ total outcomes. For two draws in a row, multiply the probability of the first draw by the probability of the second.
Without replacement, the first item stays out. The total drops by 1, and if the first item was the color you want, that count drops by 1 too. With replacement, the bag is the same for both draws.
| The question says | What to do |
|---|---|
| Without replacement, both red | Red/total × (red − 1)/(total − 1) |
| With replacement, both red | Red/total × red/total |
| The first counter is red. Probability the next is blue? | One fraction from the bag after the red is removed |
| One of each color, either order | Add the two orders |
| At least one red | 1 − probability of no red |
Multiply for draws in a row. Add only for different ways the same result can happen.
Step 2 of 6
Multiply the two draws
Worked example: Green marbles
A jar has 3 green marbles and 7 yellow marbles. Two marbles are drawn without replacement. What is the probability that both are green?
- A3/50
- B1/15
- C9/100
- D2/9
- 1
First draw
3 green out of 10: 3/10.
- 2
Update the jar
One green is gone. Now 2 green out of 9: 2/9.
- 3
Multiply
3/10 × 2/9 = 6/90 = 1/15.
- 4
See why the other choices are there
9/100 is 3/10 × 3/10, which puts the marble back. 3/50 is 3/10 × 2/10, which updates the green count but not the total. 2/9 is only the second draw.
Answer
The probability is 1/15.
Wrong: Without replacement: 3/10 × 3/10 = 9/100.
Right: Without replacement: 3/10 × 2/9 = 1/15.
After a green marble is taken out and kept, the jar holds 9 marbles and only 2 are green. The second fraction has to describe that jar.
Step 3 of 6
Either order, and at least one
For one of each, the green can come first or second, so find both orders and add them. With the same jar: 3/10 × 7/9 = 7/30 and 7/10 × 3/9 = 7/30, so one of each is 7/30 + 7/30 = 7/15.
For at least one green, it is faster to find the only way to fail, which is no green at all. Two yellows: 7/10 × 6/9 = 7/15. So at least one green is 1 − 7/15 = 8/15. Check: one of each plus both green is 7/15 + 1/15 = 8/15.
Step 4 of 6
Update, then multiply
Check yourself · Question 1
A drawer has 5 white socks and 3 black socks. Two socks are drawn at random without replacement. What is the probability that both are black?
Answer: C
First draw: 3/8. Then 2 black out of 7: 2/7. Multiply: 3/8 × 2/7 = 6/56 = 3/28.
Trap answer: 9/64
Forgetting the drawer changed
- Why it looks right
- Using 3/8 twice treats the two draws as the same, which feels fair.
- Why it's wrong
- The first black sock stays out. The second draw is from 7 socks, and only 2 are black.
- The right answer
- 3/28, because 3/8 × 2/7 = 3/28.
Check yourself · Question 2
A bag has 5 green and 3 yellow tiles. A tile is drawn, and it is yellow. It is not put back. What is the probability that the next tile is green?
Answer: B
After a yellow is removed, the bag has 5 green and 2 yellow, 7 tiles in all. Probability of green: 5/7.
Check yourself · Question 3
A bag has 3 red and 5 blue counters. Two counters are drawn without replacement. What is the probability that at least one is red?
Answer: C
No red means two blues: 5/8 × 4/7 = 20/56 = 5/14. At least one red: 1 − 5/14 = 9/14.
Common mistake
"Each draw is from the same bag, so I use the same fraction twice."
- Why it's tempting
- With replacement, that is exactly right, and the two cases look alike on paper.
- Do this instead
- Check the wording. Without replacement, subtract 1 from the total for the second draw, and subtract 1 from the color that was drawn first.
Step 5 of 6
What to remember
Remember
- For two draws in a row, multiply: first draw × second draw from the updated bag.
- Without replacement, the total drops by 1, and so does the color drawn first; with replacement, nothing changes.
- For one of each, add both orders. For at least one, use 1 − probability of none.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what fraction did you use for the second draw, and which wrong choice keeps the bag unchanged?