Determine the probability of drawing two cards from a deck( without replacement) and getting the
same suit using
1) conditional probability
2) combinations
To determine the probability of drawing two cards from a deck (without replacement) and getting the same suit, we can calculate it using both conditional probability and combinations.
1) Conditional Probability:
The conditional probability is the probability of an event occurring given that another event has already occurred. In this case, we want to calculate the probability of drawing two cards of the same suit given that the first card drawn was from a specific suit.
Let's break down the steps to calculate the probability using conditional probability:
Step 1: Determine the probability of drawing the first card from any suit.
Since there are four suits in a standard deck of cards (hearts, diamonds, clubs, and spades), the probability of drawing the first card from any suit is 1 (or 100%).
Step 2: Determine the probability of drawing the second card from the same suit as the first card.
After drawing the first card, there are 51 cards left in the deck, with 12 cards of the same suit as the first card. So, the probability of drawing the second card from the same suit is 12/51.
Step 3: Calculate the conditional probability.
The conditional probability is calculated by dividing the probability of both events occurring (drawing two cards of the same suit) by the probability of the first event occurring (drawing the first card from any suit). Thus, the conditional probability can be calculated as follows:
Conditional Probability = (Probability of both events occurring) / (Probability of the first event occurring)
Conditional Probability = (12/51) / 1
Conditional Probability = 12/51
Therefore, the probability of drawing two cards from a deck (without replacement) and getting the same suit using conditional probability is 12/51 (or approximately 0.235).
2) Combinations:
Combinations can also be used to calculate the probability of drawing two cards from a deck and getting the same suit. Let's see how to compute it using combinations:
Step 1: Determine the number of ways to choose 2 cards of the same suit.
We need to choose 2 cards of the same suit from a set of 13 cards (since each suit has 13 cards in a deck). The number of ways to choose 2 cards from 13 is given by the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of cards, and r is the number of cards chosen.
In this case, n = 13 and r = 2. Substituting these values into the combination formula:
C(13, 2) = 13! / (2!(13-2)!)
= 13! / (2!11!)
Calculating this would result in C(13, 2) = 78.
Step 2: Determine the total number of possible outcomes when choosing any 2 cards from the deck.
The total number of cards in a deck is 52, so we need to calculate the number of ways to choose 2 cards from a set of 52 cards:
C(52, 2) = 52! / (2!(52-2)!)
= 52! / (2!50!)
= 52 x 51 / 2 x 1
= 1326
Step 3: Calculate the probability using combinations.
The probability is calculated by dividing the number of favorable outcomes (i.e., choosing 2 cards of the same suit) by the total number of possible outcomes (i.e., choosing any 2 cards from the deck).
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Probability = 78 / 1326
Probability ≈ 0.0588
Therefore, the probability of drawing two cards from a deck (without replacement) and getting the same suit using combinations is approximately 0.0588 (or 5.88%).