3 gallons of paint cover 900 square feet. How many gallons will cover 300 square feet?
2 answers:
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Answer:
a)
the smallest n that maximizes the expected payoff is n=5.
b) 4
Step-by-step explanation:
a)
The expected amount of $ won for each die would be the average of 1, 2, 3, 4 and 5 which is $3.
Let W(n) the expected money won when rolling n dice.
n =1
If the gambler rolls only one die, the expected money won would be $3 times the probability of not getting a 6, which is 5/6.
So
n=2
If the gambler rolls 2 dice, the expected money won would be $3 times the probability of not getting a 6 in any of the dice. Since the outcome of the rolling does not depend on the previous rollings, the probability is
and
n=3
Similarly, since the probability of not getting a 6 in 3 dice equals
and the formula for the expected money won with n dice would be
In the picture attached there is a plot of the values of the expected money won for n=1 to 20 (See picture)
For n=5 and n=6 we get the maximum profit expected of $6.02816=$6 rounded to the nearest integer.
Hence, the smallest n that maximizes the expected payoff is n=5.
b)
The probability that face k (k=1,2,...or 6) shows up is 1/6,
as this face can be in any of the 10 positions of the arrangement, there are 10 ways that face k can show up.
The probability that face k (k=1,2,...or 6) shows up twice is
as this face can be in any of the (combinations of 10 taken 2 at a time) positions of the arrangement, there are C(10;2) ways that face k can show up twice.
The probability that face k (k=1,2,...or 6) shows up three times is
as this face can be in any of the (combinations of 10 taken 2 at a time) positions of the arrangement, there are C(10;3) ways that face k can show up twice.
So, we infer that the expectation is
and the expected number of distinct dice values that show up is 4.
The distance between given points is 3.6 units.
Step-by-step explanation:
Given points are;
(2,6) and (5,8)
Here;
Distance between two points is calculated by;
The distance between given points is 3.6 units.
Keywords: distance, points
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