Math Problem Statement
An honest coin is tossed 4 times in a row. Find the probability of each of the following events. (a) E Subscript 1: "The coin comes up heads exactly twice." (b) E Subscript 2: "All 4 tosses come up the same." (c) E Subscript 3: "Exactly one-third of the tosses come up tails." (d) E Subscript 4: "The first 2 tosses come up tails.
Solution
Let's solve the problem by breaking down each part.
We are tossing an honest (fair) coin 4 times. The outcomes of each toss are independent, and each toss has two possible outcomes: heads (H) or tails (T). The total number of possible outcomes of 4 tosses is .
(a) : "The coin comes up heads exactly twice."
To find the probability of getting exactly 2 heads in 4 tosses, we can use the binomial distribution formula. The number of ways to choose exactly 2 heads out of 4 tosses is given by the binomial coefficient:
The binomial coefficient represents the number of ways to choose 2 heads from 4 tosses:
So, the probability is:
(b) : "All 4 tosses come up the same."
For all 4 tosses to come up the same, either all 4 tosses must be heads (HHHH), or all 4 tosses must be tails (TTTT). These are the only two favorable outcomes out of 16 possible outcomes.
(c) : "Exactly one-third of the tosses come up tails."
Since we are tossing the coin 4 times, it is impossible for exactly one-third of the tosses to be tails because one-third of 4 is not a whole number (it’s ). Therefore, this event is impossible.
(d) : "The first 2 tosses come up tails."
The first 2 tosses must be tails (TT), and the last 2 tosses can be anything (either heads or tails). There are 4 possible outcomes for the last 2 tosses: HH, HT, TH, TT. Thus, there are 4 favorable outcomes out of 16 possible outcomes.
Summary of probabilities:
Would you like further clarification on any part of this?
Related Questions:
- What is the probability of getting at least one head in 4 coin tosses?
- What is the probability of getting exactly 3 tails in 4 tosses?
- How would the probabilities change if the coin were biased?
- How many possible outcomes are there for tossing the coin 5 times?
- What is the probability of getting exactly 1 head in 4 tosses?
Tip: Binomial coefficients are useful in determining the number of ways an event can happen in cases with independent trials like coin tosses.
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Math Problem Analysis
Mathematical Concepts
Probability
Binomial Distribution
Combinatorics
Formulas
Binomial coefficient: P(E1) = (4 choose 2) / 16
Basic probability formula: P(E) = favorable outcomes / total outcomes
Theorems
Binomial Theorem
Basic Probability Theorem
Suitable Grade Level
Grades 9-12