Your mom is a sporting person, so when it's close to your bedtime, she will have a contest with you to see if you get to stay up and extra half-hour to play a computer game. You get to draw a card from a well-shuffled deck. If you draw a face card, an ace, or any heart, she'll "have a heart" and let you stay up. If you draw any other card, you lose and go ahead to bed. Who has the best chance of winning, you or your Mom?
The person with the most qualifying cards in the deck has the best chance. There are 12 face cards and 4 aces. There are also 9 hearts that are not aces or face cards. You can only win with 25 cards out of 52. That is less than half the deck.
To determine who has the best chance of winning the contest, we need to analyze the probabilities of the outcomes based on the conditions given. Let's break down the possibilities step by step:
1. Probability of drawing a face card: The deck consists of 52 cards, and there are 12 face cards (Jacks, Queens, and Kings). Therefore, the probability of drawing a face card is 12/52 or 3/13.
2. Probability of drawing an ace: Since there are four aces in a deck, the probability of drawing an ace is 4/52 or 1/13.
3. Probability of drawing any heart: There are 13 hearts in a deck, so the probability of drawing a heart is 13/52 or 1/4.
Now let's calculate the chance of losing the contest:
Probability of losing = 1 - (Chance of drawing a face card + Chance of drawing an ace + Chance of drawing any heart)
= 1 - (3/13 + 1/13 + 1/4)
= 1 - (12/52 + 4/52 + 13/52)
= 1 - (29/52)
= 23/52 or 44.23%
Therefore, the chance of losing the contest and going to bed is approximately 44.23%.
To determine the chance of winning, we subtract the probability of losing from 1:
Probability of winning = 1 - Probability of losing
= 1 - 23/52
= 29/52 or 55.77%
Hence, the chance of winning the contest and getting to stay up an extra half-hour to play a computer game is approximately 55.77%.
Conclusion: Based on the probabilities, you have a slightly higher chance of winning the contest compared to your Mom.