Question 1190165
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Answers:<ul><li>A) <font color=red>Hypergeometric</font></li><li>B) <font color=red>0.93472275 approximately</font></li><li>C) <font color=red>1</font></li></ul>==================================================
Explanation for part A


The uniform distribution is where the probability of any outcome is the same. For example, rolling a single six-sided die has each side with uniform probability 1/6. 
We rule out "uniform distribution" because the probabilities aren't all the same for this jelly bean problem.


The binomial distribution can be ruled out as well. This is due to the phrasing "without replacement". 


The geometric distribution isn't used because we aren't interested in questions like "what is the probability of getting blue on the first selection? second selection? etc".


We use the hypergeometric distribution, which is effectively the binomial distribution but without replacement. 
This distribution is useful to list out the possible X values and their corresponding P(X) values. 
The X values refer to the counts of a certain color. Refer to parts B or C for more info.


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Explanation for part B


Let's compute the probability of getting all six jelly beans that aren't orange.
We have 8 blue + 7 red + 5 pink = 20 beans that aren't orange out of 8+7+5+10 = 30 jelly beans total.


The probability of getting six beans in a row that aren't orange is
P(6 not orange) = (20/30)*(19/29)*(18/28)*(17/27)*(16/26)*(15/25)
P(6 not orange) = 0.06527725
Notice the numerators counting down (20,19,...) and the denominators are counting down as well (30,29,...). This is caused by the fact the previous selection is not replaced.


Subtract this value from 1 to get
P(at least one orange) = 1 - P(6 non orange)
P(at least one orange) = 1 - 0.06527725
P(at least one orange) = 0.93472275


The events "selecting 6 non orange" and "at least one orange" are complementary events. One or the other must happen. 
Therefore the two event probabilities add to 1, which is why that previous formula is useful.


Side note: as an alternative path, you can use the formula mentioned in part C. Plug in N = 30, n = 6, K = 20, k = 6 to get the result 0.06527725 which you subtract from 1 to get the final answer for this part.


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Explanation for part C


X = number of pink selected
The set of possible X values is {0,1,2,3,4,5}


Use the hypergeometric probability formula to calculate each P(X) for the set of X values mentioned.


We'll think of each outcome as "pink" vs "not pink".
There are 5 pink and 25 non pink.<ul><li>N = population size = 30 beans total</li><li>n = sample size = 6 beans selected</li><li>K = number of pink overall in population = 5</li><li>k = number of pink selected = varies from 0 to 5 i.e. those X values mentioned earlier</li></ul>The convention is that the uppercase letters N and K go with the population counts (total beans and pink beans); while the lowercase counterparts count the sample values.


I'll show the steps on how to calculate P(X = 0)
In other words when k = 0
*[Tex \Large P(k) = \frac{C(K,k)*C(N-K,n-k)}{C(N,n)}]


*[Tex \Large P(0) = \frac{C(5,0)*C(30-5,6-0)}{C(30,6)}]


*[Tex \Large P(0) = \frac{C(5,0)*C(25,6)}{C(30,6)}]


*[Tex \Large P(0) = \frac{1*177100}{593775}]


*[Tex \Large P(0) = 0.298261] approximately


For more information, check out this link about the hypergeometric distribution
<a href = "https://online.stat.psu.edu/stat414/lesson/7/7.4">https://online.stat.psu.edu/stat414/lesson/7/7.4</a>
The notation C(n,r) refers to the nCr combination formula. Sometimes it's written in the form of *[Tex \Large {n\choose r}]


This is a useful free calculator to help check your answers
<a href = "https://stattrek.com/online-calculator/hypergeometric.aspx">https://stattrek.com/online-calculator/hypergeometric.aspx</a>


You'll follow similar steps for k = 1 through k = 5


Here's a table of the results
The P(X) values are approximate and rounded to 6 decimal places.<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td></tr><tr><td>0</td><td>0.298261</td></tr><tr><td>1</td><td>0.447392</td></tr><tr><td>2</td><td>0.213044</td></tr><tr><td>3</td><td>0.038735</td></tr><tr><td>4</td><td>0.002526</td></tr><tr><td>5</td><td>0.000042</td></tr></table>
Next, we'll multiply each X and P(X) value to form a new column<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td><td>X*P(X)</td></tr><tr><td>0</td><td>0.298261</td><td>0</td></tr><tr><td>1</td><td>0.447392</td><td>0.447392</td></tr><tr><td>2</td><td>0.213044</td><td>0.426088</td></tr><tr><td>3</td><td>0.038735</td><td>0.116205</td></tr><tr><td>4</td><td>0.002526</td><td>0.010104</td></tr><tr><td>5</td><td>0.000042</td><td>0.000210</td></tr></table>
From here you need to add up the X*P(X) results
0+0.447392+0.426088+0.116205+0.010104+0.000210 = 0.999999


That effectively rounds to 1
I suspect that the true exact answer is 1 if there wasn't rounding error (when computing the approximate P(X) values).


We expect about 1 pink jelly bean will be in the sample size of 6.
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