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Suppose you toss n coins. How many different ways can the coins land?
Solution:
Solution:
You have maybe done some coin experiments as part of a class studying probability. We know that when we toss a coin, there are two equally likely outcomes: heads (H) or tails (T). When we toss two coins, we know the options are that we could have both heads, one of each, or both tails. However, these are not equally likely. Getting one of each possibility is twice as likely as getting either both heads or both tails. Therefore, if we want to represent all the equally likely outcomes in the sample space we list HH, TH, HT, and TT. This is because we can look at these outcomes as independent events where the first toss could be one of two possibilities and the second toss can be one of two possibilities. Using the multiplication rule for counting, the total number of possible outcomes for two independent coin tosses is 2⋅2 = 4 equally likely outcomes. As a product, we can organize our solution using a table.
In the first table above we have the outcomes of the first two coin tosses; the first toss is indicated by the row and the second by the column. By pairing each row and column, we are able to generate all the possible outcomes. In the second table, we take the results of our first two tosses in each row and then pair those with each of the two possible outcomes for the third coin toss. We generate the same sixteen outcomes in the image at the top of this page in the third table where the first three tosses are represented on each row and paired with the two possible outcomes for the fourth toss. Once we have all these outcomes, we now know the sample space for a binomial coin experiment with four coins. We know the frequencies of having four coins with all four heads is 1, three heads is 4, two heads is 6, one head is 4, and no heads is 1. The relative frequencies 1/16, 4/16, 6/16, 4/16, and 1/16, therefore, are the corresponding probabilities. These are the theoretical probabilities for the binomial coin experiment with 4 coins.
In the embedded website below (https://www.randomservices.org/random/apps/BinomialCoin.html) you can simulate the binomial coin experiment. Move the slider so n = 4 and you'll see the binomial distribution that we just calculated. Pressing the "play" button will simulate the outcome of tossing n = 4 coins where heads and tails is equally likely (experiment with moving the value of p to something other than 0.50 to change the fairness of the coin). Pressing the "fast forward" button will simulate tossing 4 coins a total of 100 times before it will stop (The number after "Stop" can be changed to 10, 1000, or 10000 tosses of 4 coins), unless you press the "stop" button before the simulation finishes. You can also reset the outcomes by pressing the "back" button. As the number of times that the 4 coins are tossed increases, the empirical probabilities from the experiment will begin to approach the theoretical distribution we determined in the solution to our problem.
Finally, you can change the value of p so that each of our two outcomes are no longer equally likely. This change would allow us to consider sampling in a population of two groups, for example, where one group is a minority. If 40% of the group is the minority, then in a sample of 10, how likely is it that we would have more than half from the minority? The probability of having 6 or more in a sample of 10 being in the minority is P(Y > 5) = 16.6%. Read about other real-world scenarios here.
Can you think of probability questions that would correspond to our Trains, Towers, and Pizza scenarios?
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