The Following: The Cult of Donald Duck

Probability Level 3

60% of people in Orlando, Florida prefer Mickey Mouse to Donald Duck. If six residents of Orlando are selected at random, what is the probability that 4 or more of these people prefer Donald Duck to Mickey Mouse?

Image credit: Flickr Freshwater 2006 and Wikipedia Shaka

The answer is 0.1792.

3 solutions

Vikram Kumar
Apr 20, 2014

Getting 4 or more who prefer Donald duck means 6 people prefer Donald duck or 5 people prefer Donald duck or 4 people prefer Donald duck which gives

$\frac{4^6}{10^6} + \binom{6}{1} \frac{ 6}{10} \frac{4^5}{10^5} + \binom{6}{2} \frac{6^2}{10^2} \frac{4^4}{10^4} \approx 0.1792$

this solution is wrong. The first term you wrote must contain 10^6 not 10^5.

The answer will total up to 0.11108. Plea

vipul soni - 7 years, 1 month ago

I think Vikram solved it correctly. The exponent was a typo, the expression still evaluates correctly.

Josh Silverman Staff - 7 years, 1 month ago

I think the solution can be like this, since 60% prefer mickey mouse means 40% prefer donald duck. the question is 4 or more people prefer donald duck that is p(x>=4). here we can take p=2/5 and q=3/5. hence by using binomial distribution we may get p(x>=4)=p(x=4)+p(x=5)+p(x=6) p(x>=4)=6c4(2/5)^4(3/5)^2+6c5(2/5)^5(3/5)+6c6(2/5)^6(3/5)^0=0.896. kindly verify and reply if my above calculation is correct. note: the question is for the people prefer donald duck.

meena rajan - 7 years, 1 month ago

You've got the right argument and calculation, but I think you made a numerical mistake somehow. Here's your expression evaluated: $\approx 0.1792$

Josh Silverman Staff - 7 years, 1 month ago

solution is wrong answer is 0.54432

srinivasarao kakulapati - 7 years, 1 month ago

What's your argument?

Josh Silverman Staff - 7 years, 1 month ago

You're suggesting that a random sampling of the population has a greater than 50-50 chance of comprising more minority members than majority members? That doesn't make any sense.

Your solution of 0.54432 would be correct if the question asked "What is the probability that 4 or more of these people prefer Mickey Mouse to Donald Duck?" But it didn't, it asked for the other way around. Try flipping your 0.4's and 0.6's.

Tyler Fazzone - 7 years, 1 month ago

am i making a calculation mistake smehow i did the same thing but got 0.22176

Sayam Chakravarty - 7 years ago

Kevin Bourrillion
Apr 24, 2014

To solve this mentally, it was easier for me to imagine rolling six five-sided dice and wanting to get a 1 or 2 on at least four of them. I added up the number of possible dice rolls fitting each possibility.

All six? That would be 2^6 = 64. Five of them? That would be 2^5 * 3, times 6 ways to choose which die has the high number, = 576. Four of them? 2^4 * 3^2, times 15 ways to choose which two dice have high numbers, = 2160. Total of 2800, out of 5^6 total.

I canceled fives to get 560/5^5 then 112/5^4. I entered my answer as 112/625 then learned a decimal number is required. Well, I have 5^4 on the bottom so I must multiply both top and bottom by 2^4 = 16. This gave the answer of 0.1792.

Ali Baba
Apr 24, 2014

using formula P(X=r) = n C r $p^{r}$ $q^{n-r}$ ,

P(X=4) + P(X=5) + P(X=6) = ANSWER

The Following: The Cult of Donald Duck

Image credit: Flickr Freshwater 2006 and Wikipedia Shaka

The answer is 0.1792.

3 solutions

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