<div id="sq2.1-which-bet" class="section level4">
<h4>SQ2.1 Which bet?</h4>
<p>If you are willing to assume that birthdays in the class are independent (not a terrible assumption) and that birthdays are distributed randomly throughout the year (a terrible assumption, as it turns out!), you should take Bet #2. Here’s a way to explain why:</p>
-<p>Consider that 25 people can be combined into pairs <span class="math inline">\({25 \choose 2}\)</span> ways (which you can read
“as 25 choose 2”), which is equal to <span class="math inline">\(\frac{25 \times 24}{2} = 300\)</span> (and that little calculation is where those <a href="https://en.wikipedia.org/wiki/Binomial_coefficient">binomial coefficients</a> I mentioned in my hint come in handy).</p>
-<p>Generalizing the logic from part b of the textbook exercise last week problem, I have assumed that each of these possible pairings are independent and thus each one has a probability <span class="math inline">\(P = (\frac{364}{365})\)</span> of producing an <em>unmatched</em> set of birthdays.</p>
+<p>Consider that 25 people can be combined into pairs <span class="math inline">\({25 \choose 2}\)</span> ways (which you can read
as “25 choose 2”), which is equal to <span class="math inline">\(\frac{25 \times 24}{2} = 300\)</span> (and that little calculation is where those <a href="https://en.wikipedia.org/wiki/Binomial_coefficient">binomial coefficients</a> I mentioned in my hint come in handy).</p>
+<p>Generalizing the logic from part b of the textbook exercise last week, I have assumed that each of these possible pairings are independent and thus each one has a probability <span class="math inline">\(P = (\frac{364}{365})\)</span> of producing an <em>unmatched</em> set of birthdays.</p>
<p>Putting everything together, I can employ the multiplication rule from <em>OpenIntro</em> Ch. 3 and get the following: <span class="math display">\[P(any~match) = 1 - P(no~matches)\]</span><br />
<span class="math display">\[P(no~matches) = (\frac{364}{365})^{300}\]</span><br />
And I can let R do the arithmetic:</p>
If you are willing to assume that birthdays in the class are independent (not a terrible assumption) and that birthdays are distributed randomly throughout the year (a terrible assumption, as it turns out!), you should take Bet #2. Here's a way to explain why:
-Consider that 25 people can be combined into pairs ${25 \choose 2}$ ways (which you can read "as 25 choose 2"), which is equal to $\frac{25 \times 24}{2} = 300$ (and that little calculation is where those [binomial coefficients](https://en.wikipedia.org/wiki/Binomial_coefficient) I mentioned in my hint come in handy).
+Consider that 25 people can be combined into pairs ${25 \choose 2}$ ways (which you can read as "25 choose 2"), which is equal to $\frac{25 \times 24}{2} = 300$ (and that little calculation is where those [binomial coefficients](https://en.wikipedia.org/wiki/Binomial_coefficient) I mentioned in my hint come in handy).
-Generalizing the logic from part b of the textbook exercise last week problem, I have assumed that each of these possible pairings are independent and thus each one has a probability $P = (\frac{364}{365})$ of producing an *unmatched* set of birthdays.
+Generalizing the logic from part b of the textbook exercise last week, I have assumed that each of these possible pairings are independent and thus each one has a probability $P = (\frac{364}{365})$ of producing an *unmatched* set of birthdays.
Putting everything together, I can employ the multiplication rule from *OpenIntro* Ch. 3 and get the following:
$$P(any~match) = 1 - P(no~matches)$$
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