Probability of At Most 2 Tails and Three Heads in 4 Coin Tosses
Understanding and calculating the probability of certain outcomes in a series of coin tosses is a fundamental concept in probability theory. This article delves into the problem of determining the probability of throwing at most 2 tails when a fair coin is tossed 4 times. Additionally, we will explore the probability of obtaining three heads or two tails. These concepts are not only interesting but also have practical applications in various fields such as finance, data analysis, and game theory.
Problem Statement and Introduction to Coin Tosses
A fair coin is tossed 4 times. We need to find the probability of getting at most 2 tails or three heads. This problem involves calculating the number of favorable outcomes and then dividing it by the total number of possible outcomes.
Total Outcomes
When a fair coin is tossed 4 times, the total number of possible outcomes is calculated as follows:
(2^4 16)
Calculating Favorable Outcomes for Each Case
0 Tails, 4 Heads
There is only one way to get 4 heads (HHHH).
(1) way
1 Tail, 3 Heads
The number of ways to choose 1 tail from 4 tosses is given by the binomial coefficient (binom{4}{1}).
(binom{4}{1} 4)
The possible outcomes are: THHH, HTHH, HHTH, HHHT.
2 Tails, 2 Heads
The number of ways to choose 2 tails from 4 tosses is given by the binomial coefficient (binom{4}{2}).
(binom{4}{2} 6)
The possible outcomes are: TTHH, THTH, THHT, HTTH, HTHT, HHTT.
Total Favorable Outcomes
Adding up the number of favorable outcomes for each case:
(1 4 6 11)
Probability Calculation
The probability (P) of getting at most 2 tails is calculated as:
(P(text{at most 2 tails}) frac{11}{16})
Probability of Three Heads or Two Tails
We can also calculate the probability of obtaining three heads or two tails in 4 coin tosses. Let's break this down into two parts:
Probability of Three Heads
The number of ways to choose 3 heads from 4 tosses is given by (binom{4}{3}).
(binom{4}{3} 4)
The possible outcomes are: HHHH, HHHT, HHTH, HTHH, THHH.
Probability of Two Tails
We have already calculated the number of favorable outcomes for two tails, which is 6.
Combining the Probabilities
The probability of obtaining three heads or two tails is the sum of the probabilities of each individual event, assuming they are mutually exclusive:
(P(text{three heads or two tails}) frac{4}{16} frac{6}{16} frac{10}{16} frac{5}{8} 0.625)
Conclusion
In conclusion, the probability of getting at most 2 tails when a fair coin is tossed 4 times is (frac{11}{16}). Additionally, the probability of getting three heads or two tails is (frac{5}{8}).