One taken out, not put back
A bag has 4 red and 5 blue marbles. You take one out and keep it aside. It is red. Then you take a second one. What is the chance that the second marble is red?
- The bag has 4 red (R) and 5 blue (B) marbles. P(red) = 4/9.
- A red marble is kept out. Now 3 R and 5 B are left. That is 8 marbles. P(red) = 3/8.
- Suppose a blue marble was kept out instead. Then 4 R and 4 B are left. P(red) = 4/8 = 1/2.
A red marble is kept out. So 3 red and 5 blue marbles are left. The chance is 3/8.
The rule
Not replaced means the marble is not put back. Kept aside and kept out mean the same.
A marble kept out lowers the total by 1. It lowers its own colour by 1.
The kept marble cannot be taken again. So the bag has 1 marble fewer. It has 1 fewer of that colour.
Put the marble back, and the bag is the same again. Then P(red) stays 4/9.
| First pen kept out | Pens left | Pens that write | P(next pen writes) |
|---|---|---|---|
| writes | 11 | 7 | 7/11 |
| does not write | 11 | 8 | 8/11 |
The total is 11 in both rows. Only the kind that left is lower.
Now try it. Pick a colour. Slide to keep marbles out. Then draw the next marble many times. The draws should come near the counted chance.
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