1. ## coin flip

A biased (weighted) coin is designed so that the probability of a head on each flip is 3/5

If this biased coin is flipped until exactly 2 heads appear, what is the probability that it takes exactly 3 flips until the second head appears?

I drew a tree diagram but i dont really understand what it is asking for,

P(2H in 3 flips) = 36/125 But this doesnt make sense to me.

2. It is clear that the second head must occur on the third trial.

For the first two trials any of the following two cases are possible.
(i)H,T
(ii)T,H

Let A denote the event that exactly one head occurs in first two trials
$
P(A)=P(H,T)+P(T,H)=\frac{3}{5}.\frac{2}{5}+\frac{2 }{5}.\frac{3}{5}=\frac{12}{25}
$

Let B denote the event that head occurs on the the third trial.

$
P(B)=\frac{3}{5}
$

$
P(A\cap B)=P(A)P(B)=\frac{12}{25}.\frac{3}{5}=\frac{36}{12 5}
$

I hope it makes sense now.

3. Originally Posted by brentwoodbc
A biased (weighted) coin is designed so that the probability of a head on each flip is 3/5
This is purely a side bar.
It has been shown by two at Cal Tech that there is no biased coin.
Now that takes some qualifications.
By filliping or tossing a coin, we mean that the coin is rotating is the air and it is caught on the fly (while still in the air).
A simple thought experiment proves this. Image a coin spinning is air. Stop it.
There is an equal chance that it ends ups on either side regardless of weight.

I first heard of this in a lecture by a prominent author of a widely used textbook on probability and statistics. Her comment was: “the ink was dry on my new edition containing many ‘bias coin problems’ when this result hit the press”.

4. thats good thank you.

5. Originally Posted by Plato
This is purely a side bar.
It has been shown by two at Cal Tech that there is no biased coin.
Now that takes some qualifications.
By filliping or tossing a coin, we mean that the coin is rotating is the air and it is caught on the fly (while still in the air).
A simple thought experiment proves this. Image a coin spinning is air. Stop it.
There is an equal chance that it ends ups on either side regardless of weight.

I first heard of this in a lecture by a prominent author of a widely used textbook on probability and statistics. Her comment was: “the ink was dry on my new edition containing many ‘bias coin problems’ when this result hit the press”.
Can we have a biased dice

6. Originally Posted by Plato
This is purely a side bar.
It has been shown by two at Cal Tech that there is no biased coin.
Now that takes some qualifications.
By filliping or tossing a coin, we mean that the coin is rotating is the air and it is caught on the fly (while still in the air).
A simple thought experiment proves this. Image a coin spinning is air. Stop it.
There is an equal chance that it ends ups on either side regardless of weight.

I first heard of this in a lecture by a prominent author of a widely used textbook on probability and statistics. Her comment was: “the ink was dry on my new edition containing many ‘bias coin problems’ when this result hit the press”.
ya I guess its like if you have a 10 pound ball with a volume of .5metres and a 1 pound ball with an equal volume they fall at an equal speed, but they heavy end of things does tend to fall first