B1. At least how many times must we roll two fair dice simultaneously so that the probability of rolling the same number of pips on both dice at least once is greater than $\frac{1}{2}$?

(20 points)
B1. The power of the algebra of events when rolling two dice is $6 \cdot 6=36$. At the same time, the same number of dots can fall on both dice in 6 ways. The probability that the same number of dots falls on both dice when rolled once is therefore $p=\frac{6}{36}=\frac{1}{6}$.

Suppose we roll the dice $n$ times. Let $A$ be the event that the same number of dots falls on both dice at least once. The probability of the complementary event $\bar{A}$, that the same number of dots never falls on both dice, is $P(\bar{A})=(1-p)^{n}=\left(\frac{5}{6}\right)^{n}$. The probability of event $A$ is therefore $P(A)=1-P(\bar{A})=1-\left(\frac{5}{6}\right)^{n}$. We want this probability to be greater than $\frac{1}{2}$, so it must hold that

$$
1-\left(\frac{5}{6}\right)^{n}>\frac{1}{2}
$$

The value of the expression $1-\left(\frac{5}{6}\right)^{n}$ for $n=1,2,3,4, \ldots$ is approximately $\frac{1}{6}, \frac{11}{36}, \frac{91}{216}, \frac{671}{1296}, \ldots$ The value first exceeds $\frac{1}{2}$ at $n=4$, so we must roll the dice at least 4 times.

The finding that the power of the algebra of events is 36 ...................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-19.jpg?height=66&width=1579&top_left_y=1395&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-19.jpg?height=65&width=1579&top_left_y=1458&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-19.jpg?height=63&width=1579&top_left_y=1508&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-19.jpg?height=57&width=1579&top_left_y=1559&top_left_x=273)

The calculated value of the expression $1-\left(\frac{5}{6}\right)^{n}$ for $n=1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots . .$.

The calculated value of the expression $1-\left(\frac{5}{6}\right)^{n}$ for $n=2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-19.jpg?height=63&width=1579&top_left_y=1716&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-19.jpg?height=63&width=1576&top_left_y=1773&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_9b375d7a0e2f6834f525g-19.jpg?height=62&width=1576&top_left_y=1825&top_left_x=277)