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zlopas [31]
3 years ago
9

If two fair dice are tossed, what is the smallest number of throws, n, for which the probability of getting at least one double

6 exceeds 0.5? (note: this was one of the first problems that de méré communicated to pascal in 1654.)
Mathematics
1 answer:
amid [387]3 years ago
3 0

This is a rather famous probability problem.

The easiest way to solve this is to calculate the probability that you WON'T roll a "double 6" (or a twelve) each time you roll the dice.  There are 36 ways in which dice rols can appear and only one is a twelve.  So, for one roll, the probability that you will NOT get a twelve is (35/36)^n where 35/36 is about .97222222 and n would equal 1 for the first trial.  So for your first roll the odds that you WON'T get a 12 is .97222222.

For the second roll we calculate (35/36) to the second power or (35/36)^2 which equals about .945216.

When we get to the 24th roll we calculate (.97222222)^24 which equals 0.508596.  

For the 25th roll, we calculate (.97222222)^25 which equals  0.494468.  For the first time we have reached a probability which is lower than 50 per cent.  That is to say, after 25 rolls, we have reached a point in which the probability is less than 50 per cent that we will NOT roll a twelve.

To phrase this more clearly, after 25 rolls we reach a point where the probability is greater then 50 per cent that you will roll a 12 at least once.

Please go to this page 1728.com/puzzle3.htm and look at puzzle 48. (The last puzzle on the page).  An intersting story associated with this probability problem is that in 1952, a gambler named Fat the Butch bet someone $1,000 that he could roll a 12 after 21 throws.  (He miscalculated the odds [as we know you need 25 throws] and after several HOURS, he lost $49,000!!!)

Please go that page and it has a link to the Fat the Butch story.


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HELP: Part A: Using computer software, a correlation coefficient of r=0.51 was calculated. Based on the scatter plot, is that an
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Answer:

Part A: YES, it is.

Part B: the amount of pumpkin picked and the amount of fertilizer applied.

Step-by-step explanation:

Part A:

The closer the correlation coefficient is to 1, the stronger the relationship between two variables, and vice versa. Also, the closer the data points are on a scatter plot, the closer the correlation coefficient is to 1.

The scatter plot shown indicates a positive correlation between number of days and number of pumpkins. However, the data points are to some extent farther apart from each other. This shows a moderate relationship between the two variables. Therefore, a correlation coefficient, r, of 0.51 that was calculated can be concluded to be accurate , because an r of 0.51 depicts a moderate relationship between two variables.

Part B:

A variable that could affect the number of pumpkins picked could be amount of fertilizer applied, instead of the day in October. Thus, we can compare the amount of pumpkin picked and the amount of fertilizer applied.

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