Consider a deck of 52 cards with the following characteristics:
13 red cards, numbered 1 through 13
13 blue cards, numbered 1 through 13
13 green cards, numbered 1 through 13
13 black cards, numbered 1 through 13
1. Describe the steps necessary to calculate the theoretical probability of:
a. Drawing two red cards one after the other from the given deck if the cards are returned to the deck after each pick.
b. Drawing three cards with the same value one after the other from the given deck without replacing them in the deck after each pick.
2. Describe the steps necessary to calculate the experimental probabilities of:
a. Drawing two red cards one after the other from the given deck if the cards are returned to the deck after each pick.
b. Drawing three cards with the same value one after the other from the given deck without replacing them in the deck after each pick.
To calculate the theoretical probability of events a and b, and the experimental probabilities of events a and b using the given deck, you can follow these steps:
1. Theoretical probability of drawing two red cards one after the other with replacement:
a. Determine the total number of cards in the deck: 52.
b. Calculate the number of red cards in the deck: 13.
c. Since the cards are returned to the deck after each pick, the probability of drawing a red card on the first pick is 13/52.
d. After replacing the first card, there are still 52 cards in the deck, with 13 red cards remaining. The probability of drawing another red card on the second pick is also 13/52.
e. Multiply the probabilities of each event happening to find the probability of both events happening: (13/52) * (13/52).
2. Theoretical probability of drawing three cards with the same value one after the other without replacement:
a. Determine the total number of cards in the deck: 52.
b. Choose any value from 1 to 13 (e.g., 1: Ace, 2: Two, ..., 13: King).
c. Calculate the number of ways to choose three cards with the same value: 4 (one card from each color).
d. Calculate the number of ways to choose any three cards from the deck: Combination(52, 3) = 22,100.
e. Divide the number of ways to choose three cards with the same value by the total number of ways to choose three cards: 4 / 22,100.
3. Experimental probability of drawing two red cards one after the other with replacement:
a. Shuffle the deck thoroughly.
b. Draw a card from the deck and record whether it is red or not.
c. Replace the card in the deck.
d. Repeat steps b and c for the second card.
e. Conduct this experiment a large number of times (e.g., 1000) and count the number of times two red cards are drawn one after the other.
f. Divide the number of times two red cards are drawn by the total number of experiments (1000) to get the experimental probability.
4. Experimental probability of drawing three cards with the same value one after the other without replacement:
a. Shuffle the deck thoroughly.
b. Draw the first card from the deck and record its value.
c. Remove all other cards with the same value from the deck.
d. Repeat steps b and c for the second and third cards.
e. Conduct this experiment a large number of times (e.g., 1000) and count the number of times three cards with the same value are drawn one after the other.
f. Divide the number of times three cards with the same value are drawn by the total number of experiments (1000) to get the experimental probability.
To calculate the theoretical probability, you need to understand the total number of possible outcomes and the number of favorable outcomes for each event. Let's break it down step by step:
1a. Drawing two red cards one after the other with replacement:
- Determine the total number of cards in the deck: 52 cards.
- Calculate the number of favorable outcomes: There are 13 red cards in the deck, so the probability of drawing a red card on the first draw is 13/52. Since the cards are returned to the deck, the probability of drawing a red card on the second draw is also 13/52.
- Multiply the probabilities of each event together to calculate the theoretical probability: (13/52) * (13/52) = 169/2704 ≈ 0.0625.
1b. Drawing three cards with the same value one after the other without replacement:
- Determine the total number of cards in the deck: 52 cards.
- Calculate the number of favorable outcomes: For each card value (1 through 13), there are four cards (one in each suit) in the deck. So, the probability of drawing a card with a specific value on the first draw is 4/52. Without replacement, the probability of drawing a card with the same value on the second draw is 3/51, and on the third draw is 2/50.
- Multiply the probabilities of each event together to calculate the theoretical probability: (4/52) * (3/51) * (2/50) = 1/22100 ≈ 0.000045.
2a. To calculate the experimental probability:
- Conduct a large number of trials: Draw two cards one after the other with replacement from the given deck. Repeat this process many times (e.g., 1000 trials).
- Count the number of favorable outcomes: Count the number of trials where two red cards were drawn one after the other.
- Divide the number of favorable outcomes by the total number of trials to calculate the experimental probability.
2b. To calculate the experimental probability:
- Conduct a large number of trials: Draw three cards one after the other without replacement from the given deck. Repeat this process many times (e.g., 1000 trials).
- Count the number of favorable outcomes: Count the number of trials where three cards with the same value were drawn one after the other.
- Divide the number of favorable outcomes by the total number of trials to calculate the experimental probability.
It's important to note that while theoretical probability provides an expected value based on mathematical calculations, experimental probability is derived from actual observations and may vary due to random sampling. Conducting more trials improves the accuracy of the experimental probability.