Question

7 coins are tossed simultaneously, what is the probability of getting at least two heads? (1) \(3 / 18\) (2) \(15 / 16\) (3) \(1 / 16\) (4) \(3 / 16\)

Short answer

Answer: The probability of getting at least two heads when tossing 7 coins simultaneously is \(15/16\).

Step-by-step solution

Calculate the total number of outcomes

When we toss 7 coins, each coin has 2 possible outcomes - either a Head (H) or a Tail (T). So the total outcomes for tossing 7 coins are 2^7, which equals 128.

Find the number of successful outcomes

Since we need to find the probability of getting at least 2 heads, we need to consider the scenarios of getting 2, 3, 4, 5, 6, and 7 heads out of 7 coins. We will use the combination formula to calculate this. The combination formula is C(n, k) = n! / (k!(n-k)!), where n is the total number of items and k is the number of selected items. For 2 heads: C(7, 2) = 7! / (2!(7-2)!) = 21 For 3 heads: C(7, 3) = 7! / (3!(7-3)!) = 35 For 4 heads: C(7, 4) = 7! / (4!(7-4)!) = 35 For 5 heads: C(7, 5) = 7! / (5!(7-5)!) = 21 For 6 heads: C(7, 6) = 7! / (6!(7-6)!) = 7 For 7 heads: C(7, 7) = 7! / (7!(7-7)!) = 1 The total successful outcomes are: 21+35+35+21+7+1 = 120

Calculate the probability

Now that we have the total number of successful outcomes (120) and the total number of outcomes (128), we can calculate the probability of getting at least 2 heads by dividing the number of successful outcomes by the total number of outcomes: Probability = (Number of successful outcomes) / (Total number of outcomes) = 120/128 = 15/16 Hence, the correct answer is (2) \(15 / 16\).

Key concepts

Combination Formula

The combination formula is a mathematical way to find out how many different groups (or combinations) of items can be formed from a larger set. It's especially useful when you don't care about the order of the items in the group. The formula is denoted as \( C(n, k) \), where \( n \) is the total number of items and \( k \) is the number of items to be chosen. The formula is expressed as:\[ C(n, k) = \frac{n!}{k!(n-k)!} \]Here's a quick breakdown:- \( n! \) ("n factorial") means you multiply \( n \) by every positive whole number smaller than \( n \).- \( k! \) is the factorial of the number of selected items.- \( (n-k)! \) is the factorial of the difference between the total items and selected items.In our coin toss problem, using the combination formula allows us to compute how many different ways we can end up with a certain number of heads out of 7 coin tosses.

Coin Toss Experiment

Tossing a coin is a classic example of a probability experiment. Each toss has only two possible outcomes: Heads (H) or Tails (T). This outcome makes it easy to mathematically model since each result is equally likely. When tossing multiple coins simultaneously, the total possible outcomes can be expressed using powers of 2. Specifically, if you toss \( n \) coins, the total number of possible outcomes will be \( 2^n \). For example, tossing 7 coins results in \( 2^7 \) or 128 different outcomes. This experiment forms the basic setup for probability calculations where each outcome is treated as equally possible.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a particular outcome will occur, given a specific number of trials or experiments, each with a binary result (like our heads or tails). It's particularly useful when dealing with events that have two outcomes, such as coin tosses, where we'll either get heads or tails. The parameters for a binomial distribution are:\( n \), the number of trials (coin tosses), and \( p \), the probability of one outcome (e.g., the probability of getting heads).To solve our coin problem, we consider the probability of getting heads or tails in 7 coin tosses, treating getting at least 2 heads as our "success." The binomial distribution allows us to calculate these probabilities across multiple trials and outcomes.

Head and Tail Outcomes

When we talk about heads and tails in probability, we refer to the two potential results that can come from a single coin toss. Each has a probability of \( \frac{1}{2} \) because they are equally likely.In our exercise, we are interested in the outcome of these results across multiple tosses. Specifically, how often we get "heads" when 7 coins are tossed simultaneously. The individual outcomes can be modeled by the combination of heads and tails across the tosses.For instance, getting specifically 2 heads means understanding how many arrangements of these 2 heads and 5 tails there are, which we already calculated using the combination formula. Each specific arrangement (like HHHTTT) corresponds to a unique outcome in this setup.