Question

Suppose a coin is flipped 40 times and exactly 6 heads are rolled. Use generating functions to find the probability that no more than 5 tails are rolled in a row and there are no consecutive heads.

I'm really stuck.. and don't know how to go about this problem... a detailed solution would be much appreciated!!

Answer 1

Followings are the general ways that can be used to Approach Problems of this type:

• Start from the Least Possible Restriction and then Add units to reach the goal.
• Start from the Greatest Possible Restriction and Subtract units to reach the goal.

Here, the Least Possible Restriction will be to have 1 Tail between each of the 6 Heads, like so:

H-T-H-T-H-T-H-T-H-T-H

That is there will be a total of 11 flips.

The Greatest Possible Restriction wil be to have Exactly 5 Tails in a row everywhere you can, i.e.

TTTTT-H-TTTTT-H-TTTTT-H-TTTTT-H-TTTTT-H-TTTTT-H-TTTTT

Therefore a total of 41 flips.

As we can see from Above, the Greatest Possible Restriction is closesr to the goal of 40 coin flips,

So, we will proceed with the Greatest Possible Restriction.

Total 6 Heads are given, so we can’t remove any heads to bring us down to 40 flips.

It means,we need to remove one of the Tails.

There are a total of 7 possible places from where we could remove one of the Tails i.e. One from any bunch of the Tails.

It means, there are exactly 7 ways to satisfy the condition. Also, we have ways to arrange 40 flips with 6 Heads, therefore the probability becomes:

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