As an expert in probability and statistics, I can provide a detailed explanation of the concept of "at least" and "at most" in the context of probability, specifically when it comes to coin tosses and other similar scenarios.

When we talk about the probability of getting "at least two heads" when tossing a certain number of coins, we are essentially considering all the possible outcomes that include two or more heads. This is a way of quantifying the likelihood of an event that meets or exceeds a certain condition.

Let's take the example of tossing four coins. Each coin has two sides: heads and tails. When we toss four coins, there are \(2^4 = 16\) possible outcomes because each coin can land in two different ways. The outcomes can range from all tails (TTTT) to all heads (HHHH), with various combinations of heads and tails in between.

To calculate the probability of getting "at least two heads," we need to consider all the outcomes that have two, three, or four heads. Here's the breakdown:

1. **Two Heads (HHTT, HTHT, HTTH, THHT, THTH, TTHH):** There are six different ways to get exactly two heads. Each of these outcomes has a probability of \((1/2)^4 = 1/16\), since each coin has a 1/2 chance of landing heads or tails.


2. Three Heads (HHHT, HHTH, HTHH, THHH): There are four different ways to get exactly three heads. Each of these outcomes also has a probability of \((1/2)^4 = 1/16\).


3. Four Heads (HHHH): There is only one way to get all four heads, and the probability of this single outcome is \((1/2)^4 = 1/16\).

To find the total probability of getting "at least two heads," we add the probabilities of all these favorable outcomes:

\[ P(\text{at least two heads}) = P(\text{two heads}) + P(\text{three heads}) + P(\text{four heads}) \]
\[ P(\text{at least two heads}) = 6 \times (1/16) + 4 \times (1/16) + 1 \times (1/16) \]
\[ P(\text{at least two heads}) = (6/16) + (4/16) + (1/16) \]
\[ P(\text{at least two heads}) = 11/16 \]

So, the probability of getting at least two heads when tossing four coins is \(11/16\), not \(1/16\) as the reference content incorrectly suggests.

Now, let's discuss "at most two heads." This means we are considering the outcomes that have two heads or fewer, which includes the possibilities of zero heads (all tails), one head, and two heads. The calculation would be similar to the one above, but we would only sum the probabilities of these three sets of outcomes.

To summarize, "at least" in probability terms means "greater than or equal to," while "at most" means "less than or equal to." These concepts are crucial when calculating the likelihood of events in various scenarios, including but not limited to coin tosses.

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So the probability of at least two heads when tossing 4 coins is 1/16. Mathematically --at least-- is the same as --greater than or equal to--. But at --most two-- is the same as --less than or equal to-- So if you want at most two heads, your winning outcomes are two heads (from above = 6 winners).read more >>