A biased coin has probability 0.6 of turning up heads. You win $x if a head comes up and you lose $y if a tail comes up. If your expected winnings is $0, what is the relationship between x and y?

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AlonsoDehner AlonsoDehner · Answer 1 · 2019-12-07T13:29:30+01:00

Answer:

x,y will be in the ratio 2:3

Step-by-step explanation:

Given that a biased coin has probability 0.6 of turning up heads

In other words, number of heads we obtain by tossing the coin would be binomial with p = 0.6 and q =1-p =0.4

If you toss one time, the distribution of U, no of heads would be

U 0 1

p 0.4 0.6

Winning amount would be 0.6x and losing amount would be 0.4y

If these two are equal then we have

[tex]0.6x=0.4y\\Or x:y = 2:3[/tex]

MrRoyal MrRoyal · Answer 2 · 2022-03-03T11:41:45+01:00

The equation that represents the relationship between x and y is 6x - 4y = 0

The given parameters are:

P(Head) = 0.6

Winning = x

Lose = y

The expected value of the distribution is calculated as:

[tex]E(x) = \sum x * P(x)[/tex]

So, we have:

[tex]E(x) = x * 0.6 - (1 - 0.6) * y[/tex]

This gives

[tex]E(x) = 0.6x - 0.4y[/tex]

The expected winning is 0.

So, we have:

[tex]0.6x - 0.4y = 0[/tex]

Multiply through by 10

[tex]6x - 4y = 0[/tex]

Hence, the relationship between x and y is 6x - 4y = 0