# What is the expected value to gain in following game and how much one is expected to win if one plays #100# games?

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In a card game return on getting an ace, other than club, is #$5# but if it is a club he gets extra #$10# and getting a club, other than ace, gets you #$1# .

In a card game return on getting an ace, other than club, is

He is expected to win

in

Coming to the problem,

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To determine the expected value of the game, we need to calculate the average amount one would win (or lose) per game.

Let's denote the probability of winning 1 as \( P(\1) ), and the probability of losing 1 as \( P(-\1) ). From the given information, we know that:

( P($1) = 0.45 ) and ( P(-$1) = 0.55 )

Now, we calculate the expected value (EV) using the formula:

( EV = (P($1) \times \text{amount won}) + (P(-$1) \times \text{amount lost}) )

( EV = (0.45 \times $1) + (0.55 \times -$1) )

( EV = $0.45 - $0.55 )

( EV = -$0.10 )

This means that on average, a player can expect to lose $0.10 per game.

Now, to find out how much one is expected to win if one plays 100 games, we simply multiply the expected value per game by the number of games:

( \text{Expected winnings in 100 games} = EV \times \text{Number of games} )

( \text{Expected winnings in 100 games} = -$0.10 \times 100 )

( \text{Expected winnings in 100 games} = -$10 )

Therefore, if one plays 100 games, one is expected to win -$10 on average.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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