Probability Problem: Tossing Three Unbiased Coins
Data Science and A.I. Lecture Series
Problem Statement
Three unbiased coins are tossed simultaneously. Find the probability of:
- At least two heads
- At most two heads
- All heads
- Exactly one head
- Exactly one tail
Step 1: Sample Space
The sample space \(S\) for tossing three unbiased coins is:
\[ S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \]
The total number of outcomes is:
\[ |S| = 8 \]
Step 2: Probability of At Least Two Heads
Outcomes with two or more heads are:
\[ \{HHH, HHT, HTH, THH\} \]
The number of favorable outcomes is:
\[ 4 \]
The probability is:
\[ P(\text{At least two heads}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{4}{8} = \frac{1}{2} \]
Step 3: Probability of At Most Two Heads
Outcomes with zero, one, or two heads are:
\[ \{HTT, THT, TTH, TTT, HHT, HTH, THH\} \]
The number of favorable outcomes is:
\[ 7 \]
The probability is:
\[ P(\text{At most two heads}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{7}{8} \]
Step 4: Probability of All Heads
The only outcome with all heads is:
\[ \{HHH\} \]
The number of favorable outcomes is:
\[ 1 \]
The probability is:
\[ P(\text{All heads}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{8} \]
Step 5: Probability of Exactly One Head
Outcomes with exactly one head are:
\[ \{HTT, THT, TTH\} \]
The number of favorable outcomes is:
\[ 3 \]
The probability is:
\[ P(\text{Exactly one head}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{3}{8} \]
Step 6: Probability of Exactly One Tail
Outcomes with exactly one tail are:
\[ \{HHT, HTH, THH\} \]
The number of favorable outcomes is:
\[ 3 \]
The probability is:
\[ P(\text{Exactly one tail}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{3}{8} \]
Final Results
- Probability of at least two heads: \( \frac{1}{2} \)
- Probability of at most two heads: \( \frac{7}{8} \)
- Probability of all heads: \( \frac{1}{8} \)
- Probability of exactly one head: \( \frac{3}{8} \)
- Probability of exactly one tail: \( \frac{3}{8} \)
PDF Presentation
Tossing Three Unbiased CoinsVideo
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