Problem 55 - Algebra 2

Question

Find each probability if a coin is tossed 4 times \(P(\text { at least } 1 \text { head })\)

Step-by-Step Solution

Answer

The probability of getting at least 1 head is \( \frac{15}{16} \).

Step 1: Understand the Total Possible Outcomes

When a coin is tossed 4 times, each toss has 2 possible outcomes: heads (H) or tails (T). Thus, the total number of possible outcomes when a coin is tossed 4 times is calculated as: \[ 2^4 = 16 \]

Step 2: Calculate the Probability of Getting No Heads

The only way to have no heads in 4 tosses is to get tails on all 4 tosses (TTTT). This is only 1 specific outcome. Hence, the probability of getting no heads is:\[ P(\text{no heads}) = \frac{1}{16} \]

Step 3: Use Complementary Probability

To find the probability of getting at least 1 head, use the complement of the event of getting no heads. This can be calculated as:\[ P(\text{at least 1 head}) = 1 - P(\text{no heads}) \]Substitute the known value:\[ P(\text{at least 1 head}) = 1 - \frac{1}{16} = \frac{15}{16} \]

Key Concepts

Complementary Probability

Sometimes, calculating the probability of an event happening directly can be tricky. That's where complementary probability comes in. When dealing with probabilities, there's a helpful rule: the probability of an event and its complement, the opposite event, must add up to 1.
Let's look at our example. To find the probability of getting at least one head when tossing a coin 4 times, it was easier to first find the probability of the opposite event – getting no heads. This is because there's only one way to get all tails: TTTT.
The probability of no heads (all tails) is \[ P(\text{no heads}) = \frac{1}{16} \]. Given this, the probability of the original event, at least one head, is calculated via the complement rule:

• Find the probability of the opposite event.
• Subtract that probability from 1 to get the desired probability.

So, \[ P(\text{at least 1 head}) = 1 - P(\text{no heads}) = 1 - \frac{1}{16} = \frac{15}{16} \]. Using complementary probabilities simplifies complex calculations, ensuring easier results.

Total Possible Outcomes

Understanding the concept of total possible outcomes is key in probability.In probability, you often start by considering all possible outcomes of an action.
For a simple coin toss, you have two outcomes: heads (H) or tails (T).
Now, when you toss a coin 4 times, you can think of this as a sequence of independent events. Each toss is separate, and each has two possible results, which we call binary outcomes: H or T.
To find the total number of outcomes when tossing a coin multiple times, you raise the number of outcomes per event to the power of the number of events. Hence, for 4 coin tosses, the total possible outcomes are calculated as:

• \[ 2^4 = 16 \] outcomes.

These 16 outcomes include every possible combination of heads and tails for the 4 tosses.

Heads and Tails

When flipping a coin, recognizing heads and tails as possible outcomes is fundamental. Each coin toss gives precisely one of two outcomes: heads (H) or tails (T). Let's break this further:
Each toss is independent, meaning the result of one does not affect the next. When you toss the coin 4 times, you essentially perform this independent action repeatedly, with each maintaining the outcome probability of 1/2 for heads and 1/2 for tails.
Visualize it this way:

• 1st toss: 2 options (H or T)
• 2nd toss: 4 possible combinations (HH, HT, TH, TT)
• 3rd toss: 8 combinations, etc.

The sequence of heads and tails grows exponentially, leading to 16 possible unique sequences after 4 flips, as calculated by \[ 2^4 \]. This understanding of binary outcomes makes tackling more complex probability questions achievable through simple step-by-step deductions.