suppose a student is offered a chance to draw a card from ordinary deck of 52 playing cards and win $10 if he or she draws an ace, $ 2 for a King, and $1 for a queen. If $2 must be paid to play the game, what is the expected winnings every time the game is played by the students?
Ah, the world of gambling and probabilities! Let's crunch some numbers and discover the expected winnings for our ambitious student.
To start, let's determine the probabilities of drawing each card:
1. The deck has 52 cards, and there are 4 aces. So, the probability of drawing an ace is 4/52 or 1/13.
2. Similarly, there are 4 kings, so the probability of drawing a king is also 4/52 or 1/13.
3. Finally, there are 4 queens, resulting in the probability of drawing a queen being 4/52 or 1/13.
Now, let's calculate the expected winnings by multiplying the probability of drawing each card by its respective payout:
Expected winnings = (Probability of drawing an ace) * ($10) + (Probability of drawing a king) * ($2) + (Probability of drawing a queen) * ($1) - ($2)
Expected winnings = (1/13) * ($10) + (1/13) * ($2) + (1/13) * ($1) - ($2)
Expected winnings = ($10/13) + ($2/13) + ($1/13) - ($2)
Calculating this gives us an expected winnings of $10/13 + $2/13 + $1/13 - $2 = $0
So, the expected winnings every time this game is played is $0.
However, speaking of unexpected winnings, don't forget that happiness and laughter can be priceless! Keep that in mind, my friend.
To calculate the expected winnings every time the game is played, we can multiply the probability of winning each amount by the corresponding winnings and sum them up.
First, let's determine the probability of drawing each card:
- There are 4 aces in a deck of 52 cards, so the probability of drawing an ace is 4/52 = 1/13.
- There are 4 kings, so the probability of drawing a King is also 4/52 = 1/13.
- Similarly, there are also 4 queens, so the probability of drawing a Queen is 4/52 = 1/13.
- The remaining cards (queens, jacks, 10s, 9s, 8s, 7s, 6s, 5s, 4s, 3s, and 2s) do not win any prize.
Next, let's calculate the expected winnings for each card:
- The winnings for drawing an Ace is $10.
- The winnings for drawing a King is $2.
- The winnings for drawing a Queen is $1.
Finally, let's calculate the expected winnings:
Expected winnings = (Probability of drawing an Ace × Winnings for an Ace) + (Probability of drawing a King × Winnings for a King) + (Probability of drawing a Queen × Winnings for a Queen) - Cost to play the game.
Expected winnings = (1/13 × $10) + (1/13 × $2) + (1/13 × $1) - $2
Expected winnings = $0.7692 + $0.1538 + $0.0769 - $2
Expected winnings ≈ -$0.9999
Therefore, the expected winnings every time the game is played by the students is approximately -$0.9999. This means that on average, the student is expected to lose about $0.9999 for each game played.
To calculate the expected winnings of the game, we need to calculate the probability of drawing each card and multiply it by the corresponding amount of money won.
Let's start by calculating the probabilities of drawing each card:
1. There are 4 aces in a deck of 52 cards, so the probability of drawing an ace is 4/52.
2. There are 4 kings, so the probability of drawing a king is also 4/52.
3. There are 4 queens, so the probability of drawing a queen is 4/52.
Now let's calculate the expected winnings:
Expected winnings = (Probability of drawing an ace * Amount won for an ace) + (Probability of drawing a king * Amount won for a king) + (Probability of drawing a queen * Amount won for a queen) - Cost to play the game
Expected winnings = (4/52 * $10) + (4/52 * $2) + (4/52 * $1) - $2
Expected winnings = $0.7692 + $0.1538 + $0.0769 - $2
Expected winnings = $-0.7692
Therefore, the expected winnings every time the game is played by the students is -$0.7692 or approximately -$0.77. This means that, on average, the student can expect to lose about $0.77 each time they play the game.
10*1/13 + 2*1/13 + 1*1/13 - 2 = -1
better go back to class and stop gambling