You pick a card at random from an ordinary deck of 52 cards. If the card is an ace, you get 9 points; if not, you lose 1 point.
Write the equation for the expected value.
[tex]\begin{aligned}
E(V) & =\frac{4}{52}(a)+\frac{b}{52}(c) \\
a & =\square c=\square b=\square c=\square
\end{aligned}[/tex]
Answer (1)
The probability of drawing an ace is 52 4 , and you get 9 points, so a = 9 .
The probability of not drawing an ace is 52 48 , and you lose 1 point, so c = − 1 and b = 48 .
The equation for the expected value is E ( V ) = 52 4 ( 9 ) + 52 48 ( − 1 ) .
Therefore, a = 9 , c = − 1 , and b = 48 .
a = 9 , c = − 1 , b = 48
Explanation
Analyze the problem and data We are given a scenario where we pick a card from a standard 52-card deck. If we draw an ace, we gain 9 points, and if we don't, we lose 1 point. Our goal is to express the expected value E(V) in the form:
E ( V ) = 52 4 ( a ) + 52 b ( c )
We need to determine the values of a , b , and c .
Determine the value of a The probability of drawing an ace from the deck is 52 4 , since there are 4 aces in a 52-card deck. If we draw an ace, we get 9 points. Therefore, a = 9 .
Determine the values of b and c The probability of not drawing an ace is the number of non-ace cards divided by the total number of cards. There are 52 − 4 = 48 non-ace cards. So the probability of not drawing an ace is 52 48 . If we don't draw an ace, we lose 1 point, which means we get -1 points. Therefore, c = − 1 and b = 48 .
Write the complete equation Now we can write the complete equation for the expected value:
E ( V ) = 52 4 ( 9 ) + 52 48 ( − 1 )
State the final values So, we have a = 9 , b = 48 , and c = − 1 .
Examples
This type of expected value calculation is used in many real-world scenarios, such as in the insurance industry to calculate premiums. For example, an insurance company might calculate the expected payout for a policy based on the probability of certain events occurring. Similarly, in finance, expected value is used to assess the potential profitability of investments, considering both the potential gains and the risks involved. In games of chance, like lotteries or casino games, understanding expected value helps players assess whether the game is favorable to them in the long run. For instance, if the expected value of a lottery ticket is negative, it means that, on average, a player will lose money for each ticket they buy.
