02 / 12 · Count the chances
Does the first draw change the second?
Notice whether a draw changes the next chance before multiplying probabilities.
Builds on One out of four
Go to practice ↓A question to keep in mind
Two stars and two plain cards: does putting a card back matter?
If you return the card and mix well, the next draw still uses the same four cards. Its chance stays the same. We call the draws independent when learning the first result does not change the next chance.
If you keep a star card out, only one star remains among three cards. Now the next star chance is 1/3, not 1/2. For two stars, multiply the first chance by the second chance given that the first was a star.
Work through an example
- With replacement: first star chance = 2/4.
- After a star is returned: second star chance = 2/4 again.
- Two stars: 2/4 x 2/4 = 1/4 = 25%.
Your turn
0 / 3Return and mix after each draw. What is the chance of a star on the first draw, and of stars on both draws?
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- First draw is a star: 50 %
- Both draws are stars: 25 %
The first chance is 50%. Both stars requires two successes: 0.5 x 0.5 = 0.25, or 25%.
You drew a star and did not return it. What is the chance that the next card is a star?
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1/3
The second chance is about the remaining box, not the original box.
A fair coin is tossed independently each time. After five heads, what is the chance of tails next?
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Still 50%.
We are told the coin is fair and the tosses independent. Under those assumptions, each new toss still has a 50% tails chance.
Bring it back to your own work
Write one example with replacement and one without it. Explain in words why the second chance stays the same or changes.
Statistical literacy: directory, audit and transfer assessment
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