Probability Analysis: All Three Cards Are Hearts Given At Least Two Hearts
" "When drawing three cards from a standard 52-card deck without replacement, we are often interested in specific outcomes and their probabilities. In this article, we will explore the conditional probability of drawing three hearts given that at least two of the three cards are hearts. This will involve several steps, including the use of combinatorial methods and conditional probability rules.
" "Step-by-Step Solution
" "Let's denote:
" "( A ): the event that all three cards are hearts.
" "( B ): the event that at least two of the three cards are hearts.
" "We want to find ( P(A|B) ), the probability that all three cards are hearts given that at least two of the three cards are hearts. This can be calculated using the formula for conditional probability:
" "Conditional Probability Formula: ( P(A|B) frac{P(A cap B)}{P(B)} )
" "Step 1: Calculate ( P(A cap B) )
" "The event ( A ) (all three cards are hearts) is a subset of event ( B ) (at least two cards are hearts). Therefore:
" "( P(A cap B) P(A) )
" "To find ( P(A) ), we calculate the probability of drawing 3 hearts from the 13 hearts in the deck. This can be done using combinations:
" "Number of ways to choose 3 hearts from 13 hearts: ( binom{13}{3} frac{13 times 12 times 11}{3 times 2 times 1} 286 )
" "Total ways to choose 3 cards from 52 cards: ( binom{52}{3} frac{52 times 51 times 50}{3 times 2 times 1} 22100 )
" "Thus, ( P(A) frac{286}{22100} ).
" "Step 2: Calculate ( P(B) )
" "Next, we need to calculate ( P(B) ), the probability of drawing at least 2 hearts. This can happen in two scenarios:
" "" "Exactly 2 hearts and 1 non-heart" "Exactly 3 hearts (which is the same as scenario 1)" "" "1. Probability of exactly 2 hearts and 1 non-heart:
" "Number of ways to choose 2 hearts from 13 hearts: ( binom{13}{2} frac{13 times 12}{2 times 1} 78 )
" "Number of ways to choose 1 non-heart from 39 cards: ( binom{39}{1} 39 )
" "So, the total ways to choose 2 hearts and 1 non-heart:
" "( 78 times 39 3042 )
" "2. Probability of exactly 3 hearts (which is the same as the previous calculation):
" "( 286 )
" "Now we combine these probabilities:
" "Total ways to choose at least 2 hearts: ( frac{3042 286}{22100} frac{3328}{22100} )
" "Therefore, ( P(B) frac{3328}{22100} ).
" "Step 3: Calculate ( P(A|B) )
" "Using the conditional probability formula:
" "( P(A|B) frac{P(A cap B)}{P(B)} frac{frac{286}{22100}}{frac{3328}{22100}} frac{286}{3328} )
" "Step 4: Simplify the fraction
" "To simplify ( frac{286}{3328} ), we find the greatest common divisor (GCD) of 286 and 3328, which is 2:
" "( frac{286 div 2}{3328 div 2} frac{143}{1664} )
" "Final Answer
" "The probability that all three cards are hearts given that at least two of the three cards are hearts is:
" "( frac{143}{1664} )
" "Conclusion
" "Through a step-by-step analysis and application of combinatorial methods and conditional probability, we have calculated the probability that all three cards are hearts given that at least two of the three cards are hearts. Understanding such concepts is crucial for solving probability problems in various fields, including statistics, data analysis, and game theory.