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Chapter 10: Problem 10

Find the probability of the given event. Obtaining exactly 1 head when 3 coins are tossed

Short Answer

Expert verified
The probability is \(\frac{3}{8}\).

Step by step solution

01

Determine Total Number of Outcomes

Each coin can land as Heads (H) or Tails (T). Therefore, for 3 coins tossed, the total number of possible outcomes is \(2^3 = 8\). These outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
02

Identify Favorable Outcomes

To find the instances where exactly 1 head appears, we list the outcomes with only one 'H': HTT, THT, TTH. These are the favorable outcomes.
03

Calculate Probability

The probability is given by the ratio of the number of favorable outcomes to the total number of outcomes. Here, there are 3 favorable outcomes (HTT, THT, TTH) and 8 total outcomes. Thus, the probability is \(\frac{3}{8}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coin Toss
A coin toss is a classic probability exercise because each coin has two possible outcomes: Heads (H) or Tails (T). When you toss a coin, these represent all potential outcomes. Tossing one coin is simple, but when you toss multiple coins, the number of possible outcomes multiplies.
For example, with 3 coins, each one has the possibility of Heads or Tails. To find how many possible combinations there are, you calculate it as follows:
  • The first coin can be H or T.
  • The second coin can also be H or T.
  • The third coin, again, can be H or T.
Using the concept of multiplication, you calculate the total outcomes using the formula for combinations, which here is simplified as: \[2^3 = 8\] So, there are 8 unique sequences possible when tossing 3 coins.
Favorable Outcomes
Favorable outcomes are the specific results you're looking for within all the possible outcomes. For a clearer picture, let's focus on our example where we want exactly one head when tossing 3 coins.
To determine this, list out each possible result of the toss:
  • HHH
  • HHT
  • HTH
  • HTT
  • THH
  • THT
  • TTH
  • TTT
From these outcomes, we extract only the ones with exactly one "H":
  • HTT
  • THT
  • TTH
These are the favorable outcomes. We know that having only one head among three coins can occur in exactly three different ways, highlighted above.
Total Outcomes
Total outcomes refer to all the possible sequences or results you could get from an event, without considering any specific conditions. In our 3-coin toss example, we determine all potential results each time the coins are tossed.
Using our formula, the total outcomes equation was: \[2^3 = 8\] Here, the exponent 3 represents the number of coins, and the base 2 signifies the heads and tails options for each coin. This gives us a holistic view of every possible result:
  • HHH
  • HHT
  • HTH
  • HTT
  • THH
  • THT
  • TTH
  • TTT
These are all the sequences that can occur, summing up to a total of 8, underpinning the larger idea of probability where each combination is counted first before determining specific conditional outcomes.

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Most popular questions from this chapter

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