[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 563
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What is the probability that when three dice are rolled simultaneously, 2 points will appear on 2 dice?
You mean 2 on one and 2 on the second, and on the third no matter what? Or on the third one, is it necessarily not 2?
What is the probability that when three dice are rolled simultaneously, 2 points will appear on 2 dice?
Here is my solution:
Denote the events: A = "2 points appear on the first dice"B = "2 points on the second dice"
C = "2 points on the third dice"
The sought event X is described by the following combination:
Since events A, B and C are incompatible and independent, the probability of event X is determined by the formula:
P(X) = 0.17 ? 0,17 ? 0,83 + 0,83 ? 0,17 ? 0,17 + 0,17 ? 0,83 ? 0,17 = 0,17 ? 0,17 ? 0,83 ? 3 = 0,07.
ANSWER: The probability that 2 points will appear on 2 dice when three dice are rolled simultaneously is 0.07.
And here's another one. A very funny one in my opinion.
A die is rolled twice.
Draw the distribution law of a random variable X - the number of twos.
Find the mathematical expectation and variance of the random variable.
2) Let us find the probability of event A = "When the dice are rolled, a deuce fell out". To calculate the probability of occurrence of this event, we will use the classical definition of event probability, according to which the probability is determined by the formula
where m is the number of outcomes in which the event A appears, n is the total number of elementary incompatible equally possible outcomes.
In our case m = 1, and n = 6 (as there are six faces with numbers on the dice).
Then
3) Let us use Bernoulli's formula to determine the probability that a deuce will fall 0, 1 or 2 times:
4) Find the probability that the two on the dice will not fall out once (X=0).
5) Find the probability that the two on the dice will fall once (X=1).
6) Find the probability that the two on the dice will fall twice (X=2).
7) Let us now fill in the table expressing the distribution law of a random variable X:
.
8) Let us define the mathematical expectation of a given random variable X (the mathematical expectation describes the mean value of a random variable in a large number of trials):
9) Find the variance for a given random variable using the formula (the variance describes the mean square of the deviation of a random variable from the mean):
10) Define the standard deviation, which characterises the mean deviation of a random variable from the mean, by the formula:
ANSWER: The mathematical expectation of a random variable is M(X) = 0.334. The variance of a random variable is D(X) = 0.278.
Here is my solution:
Denote the events: A = "2 points appear on the first dice"B = "2 points on the second dice"
C = "2 points on the third dice"
The sought event X is described by the following combination:
Since events A, B and C are incompatible and independent, the probability of event X is determined by the formula:
P(X) = 0.17 ? 0,17 ? 0,83 + 0,83 ? 0,17 ? 0,17 + 0,17 ? 0,83 ? 0,17 = 0,17 ? 0,17 ? 0,83 ? 3 = 0,07.
ANSWER: The probability that 2 points will appear on 2 dice when three dice are rolled simultaneously is 0.07.
This solution is exactly the same as the previous one.
2x^2+3x-5=0
x=?
the solution is ridiculously simple - so...
2x^2+3x-5=0
x=?
the solution is ridiculously simple - so...
2x^2+3x-5=0
x=?
the solution is ridiculously simple - so...
Hidden advertising again, you're at it again .
Hidden advertising again, there you go again.