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A fair die has not shown a six in its last 10 rolls…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A fair die has not shown a six in its last rolls. A player says a six is now due. Find the probability that the next roll is a six. Then find the probability that the next rolls show no six, and the probability that they show at least one six. Give four decimal places.
Need a reminder? The key ideaShow
For independent tries the past does not change the next result. Every sequence of fair tosses has probability . Long streaks are normal in long sequences.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The next roll. , the same as always. A six is not due.
Step 2. No six in 4 rolls. .
Step 3. At least one. .
Answer: about , and .
Watch out. Ten rolls with no six is not even unusual. It happens with probability , about one time in six.
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