Probability of Choosing Chocobars and Ice Creams in Sequential Order
" "Tom has a box containing 4 chocobars and 4 ice creams. He selectively eats 3 of them one after another. What is the probability of him sequentially choosing 2 chocobars followed by an ice cream?
" "Understanding the Problem
" "Let's denote the chocobars as 'C' and the ice creams as 'I'. Tom needs to eat 3 out of these 8 items in such a way that 2 are chocobars and 1 is an ice cream. We need to determine the probability of this desirable outcome.
" "Total Number of Ways to Choose 3 Items
" "The total number of ways to choose any 3 items from the 8 available can be calculated using the combination formula:
" "[ binom{8}{3} frac{8!}{3!(8-3)!} frac{8 times 7 times 6}{3 times 2 times 1} 56 ]
" "Number of Favorable Outcomes (2 Chocobars and 1 Ice Cream)
" "First, let's calculate the number of ways to choose 2 chocobars out of 4:
" "[ binom{4}{2} frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6 ]
" "Next, let's calculate the number of ways to choose 1 ice cream out of 4:
" "[ binom{4}{1} frac{4!}{1!(4-1)!} frac{4}{1} 4 ]
" "The total number of favorable outcomes is the product of these two combinations:
" "[ 6 times 4 24 ]
" "Calculating the Probability
" "Now, the probability of Tom sequentially choosing 2 chocobars and 1 ice cream is the ratio of favorable outcomes to the total number of outcomes:
" "[ P frac{24}{56} frac{3}{7} ]
" "Thus, the probability of Tom sequentially choosing 2 chocobars and 1 ice cream is (frac{3}{7}).
" "Application in Real Life
" "This problem is not just a theoretical exercise but can be applied in various real-life scenarios such as game strategies, customer preference analysis, and more.
" "Key Combinatorial Formulae
" "" "[ binom{n}{k} frac{n!}{k!(n-k)!} ]" "[ binom{a}{b} times binom{c}{d} text{Favorable outcomes} ]" "[ P frac{text{Favorable outcomes}}{text{Total outcomes}} ]" "" "Conclusion
" "In conclusion, by following the steps of combinatorial mathematics, we can determine the probability of a specific outcome in a seemingly simple problem. This example demonstrates the application of combinatorics in everyday problem-solving.
" "For more complex probability problems and detailed analysis, consider exploring further topics in combinatorics and statistics.