The exact probability of getting exactly 2 heads when tossing 4 coins simultaneously is 3/8, which is equivalent to 37.5%.
Understanding the Basics of Coin Toss Probability
When calculating probabilities for coin tosses, it's essential to consider all possible outcomes. Each coin toss is an independent event with two potential outcomes: Heads (H) or Tails (T). When multiple coins are tossed simultaneously, the total number of possible outcomes increases exponentially.For four "fair" coins, each toss has 2 outcomes. Therefore, the total number of equally likely outcomes is calculated by raising the number of outcomes per coin (2) to the power of the number of coins (4).
- Total Outcomes = 24 = 16
These 16 outcomes represent all the unique combinations of heads and tails possible when tossing four coins.
Identifying Favorable Outcomes (Exactly 2 Heads)
To find the probability of getting exactly 2 heads, we need to determine how many of these 16 total outcomes feature precisely two heads and two tails. These are known as the favorable outcomes.There are 6 specific combinations that result in exactly 2 heads:
- HHTT (Heads, Heads, Tails, Tails)
- HTHT (Heads, Tails, Heads, Tails)
- HTTH (Heads, Tails, Tails, Heads)
- THHT (Tails, Heads, Heads, Tails)
- THTH (Tails, Heads, Tails, Heads)
- TTHH (Tails, Tails, Heads, Heads)
These combinations can also be determined using the binomial coefficient formula, often written as "n choose k" or C(n, k), where 'n' is the total number of trials (coins) and 'k' is the number of successful outcomes (heads). For 4 coins and 2 heads, this is C(4, 2) = 4! / (2! (4-2)!) = 24 / (2 2) = 6.
Calculating the Probability
Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.- Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
Applying the values we've determined:
- Probability = 6 / 16
This fraction can be simplified:
- Probability = 3 / 8
To express this as a percentage, multiply the decimal equivalent by 100:
- Probability = (3 / 8) × 100% = 0.375 × 100% = 37.5%
Summary of Probability Calculation
The calculation can be summarized as follows:| Component | Value |
|---|---|
| Total Possible Outcomes | 16 |
| Favorable Outcomes (2 Heads) | 6 |
| Probability (Fraction) | 6/16 = 3/8 |
| Probability (Percentage) | 37.5% |
Understanding basic probability concepts, such as identifying total outcomes and specific favorable events, is crucial for analyzing chance in various scenarios. For more on probability, you can explore resources like Wikipedia's article on Probability Theory.
