Question

In the final round of a TV game show, contestants have a chance to increase their current winnings of $1 million to $2 million. If they are wrong, their prize is decreased to $500,000. A contestant thinks his guess will be right 50% of the time. Should he play? What is the lowest probability of a correct guess that would make playing profitable?

They should play because the payoff will be positive.

.50 X 1,00,000 = 500,000 Total gain if you win

.50 X 500,000 = 250,000 Total loss if you lose

.5 X 1,000,000 + .5 X -500,000= 250,000

So the expected value of playing 1 million is $1,250,000

p1,000,000 + (-500,000) + p500,000

=p1,500,000 - 5000,000

500,000 =    p1,500,000

1,500,000     1,500,000

⅓ or 33%

I can do the math once given the equation but I am not understanding why I should use the equation.

Answer 1

If Contestants Wins,

They will earn $ 1,000,000 and If They take wrong decision,

They will loose $500, 000.

Probability of Winning And Loosing is 0.50 Therefore,

0.50 X (+ 1000,000) =$500, 000

0.50 X (- 500,000) =$250, 000

While, Taking in to account the probability of winning and losing,

We can see that Contestants have 50% chance of winning $ 500,000 and 50% chance of losing $250,000. So, Contestants should play. If probability of loss is greater than the probability of winning then, Contestants should not play.

If probability of winning will be 25% then,

Contestants earn,

0.25 X (+ 1000,000) = $ 250,000 which is equal to the losing of amount.

So, Lowest Probability of a correct guess that make Contestants profitable is 26%.

When Probability is 26% then participants wins: 0.26 X(+ 1000,000) =$260, 000