Let $S = \left\{ w _ { 1 } , w _ { 2 } , \ldots \right\}$ be the sample space associated to a random experiment. Let $P \left( w _ { n } \right) = \frac { P \left( w _ { n - 1 } \right) } { 2 } , n \geq 2$ . Let $A = \{ 2 k + 3 l ; k , l \in \mathbb { N } \}$ and $B = \left\{ w _ { n } ; n \in A \right\}$. Then $P ( B )$ is equal to (1) $\frac { 3 } { 32 }$ (2) $\frac { 3 } { 64 }$ (3) $\frac { 1 } { 16 }$ (4) $\frac { 1 } { 32 }$

There are nine cards on which the integers from 1 to 9 are written in a box. Two cards are taken simultaneously from this box. Let $S$ denote the sum of the numbers written on the two cards.
(1) The probability that $S$ is 5 or less is $\frac { \mathbf { A } } { \mathbf { B } }$. Let us assign a score to the result $S$.
When $S$ is 5 or less the score is $10 - S$, and when it is greater than 5 the score is 2. Then the expected value of the score is $\frac { \mathbf { C D } } { \mathbf { E F } }$.
(2) Let us perform the above trial twice, returning the two cards to the box before the second trial.
(i) The probability that $S$ is 5 or less in both trials is $\frac { \mathbf { G } } { \mathbf { H } }$.
(ii) The probability that $S$ is 5 or less in at least one trial is $\frac { \mathbf { J K } } { \mathbf { L M } }$.

Q2 Consider dice-X with the following property: when it is rolled, for numbers 1 through 5 the probabilities of that number's coming up are all the same, but the probability that 6 comes up is twice that of any other number.
(1) Denote by $p$ the probability that a particular number 1 through 5 comes up. The probability that number 6 comes up is $\mathbf { L }$ p. Since the probability of the whole event is $\mathbf { M }$, we have $p = \frac { \mathbf { N } } { \mathbf { O } }$.
(2) Dice-X is rolled twice in succession. Let us denote by $A$ the event that for both rolls the number that comes up is 1 through 5, and by $B$ the event that number 6 comes up at least once. Then the probability of event $A$, $P(A)$, and that of event $B$, $P(B)$, are
$$P ( A ) = \frac { \mathbf { PQ } } { \mathbf { RS } } , \quad P ( B ) = \frac { \mathbf { TU } } { \mathbf { VW } } .$$
Hence, we see that $\mathbf { X }$. (For $\mathbf { X }$, choose the correct answer from among choices (0) $\sim$ (4) below.) (0) $P ( A )$ is less than $P ( B )$ and the difference between them is not less than $\frac { 1 } { 36 }$.
(1) $P ( A )$ is less than $P ( B )$ and the difference between them is less than $\frac { 1 } { 36 }$.
(2) $P ( A )$ and $P ( B )$ are the same.
(3) $P ( A )$ is greater than $P ( B )$ and the difference between them is not less than $\frac { 1 } { 36 }$.
(4) $P ( A )$ is greater than $P ( B )$ and the difference between them is less than $\frac { 1 } { 36 }$.
(3) Next, dice-X is rolled three times in succession. Let us denote by $C$ the event that for all three rolls the number which comes up is 1 through 5, and by $D$ the event that the number 6 comes up at least once. When the probability $P(C)$ is compared with the probability $P(D)$, we see that $\mathbf { Y }$. (For $\mathbf { Y }$, choose the correct answer from among choices (0) $\sim$ (4) below.) (0) $P ( C )$ is less than $P ( D )$ and $P ( D )$ is not less than twice $P ( C )$.
(1) $P ( C )$ is less than $P ( D )$ and $P ( D )$ is less than twice $P ( C )$.
(2) $P ( C )$ and $P ( D )$ are the same.
(3) $P ( C )$ is greater than $P ( D )$ and $P ( C )$ is not less than twice $P ( D )$.
(4) $P ( C )$ is greater than $P ( D )$ and $P ( C )$ is less than twice $P ( D )$.

(Course 2) Q2 Consider dice-X with the following property: when it is rolled, for numbers 1 through 5 the probabilities of that number's coming up are all the same, but the probability that 6 comes up is twice that of any other number.
(1) Denote by $p$ the probability that a particular number 1 through 5 comes up. The probability that number 6 comes up is $\mathbf { L }$ p. Since the probability of the whole event is $\mathbf { M }$, we have $p = \frac { \mathbf { N } } { \mathbf { O } }$.
(2) Dice-X is rolled twice in succession. Let us denote by $A$ the event that for both rolls the number that comes up is 1 through 5, and by $B$ the event that number 6 comes up at least once. Then the probability of event $A$, $P(A)$, and that of event $B$, $P(B)$, are
$$P ( A ) = \frac { \mathbf { PQ } } { \mathbf { RS } } , \quad P ( B ) = \frac { \mathbf { TU } } { \mathbf { VW } }$$
Hence, we see that $\mathbf { X }$. (For $\mathbf { X }$, choose the correct answer from among choices (0) $\sim$ (4) below.) (0) $P ( A )$ is less than $P ( B )$ and the difference between them is not less than $\frac { 1 } { 36 }$.
(1) $P ( A )$ is less than $P ( B )$ and the difference between them is less than $\frac { 1 } { 36 }$.
(2) $P ( A )$ and $P ( B )$ are the same.
(3) $P ( A )$ is greater than $P ( B )$ and the difference between them is not less than $\frac { 1 } { 36 }$.
(4) $P ( A )$ is greater than $P ( B )$ and the difference between them is less than $\frac { 1 } { 36 }$.
(3) Next, dice-X is rolled three times in succession. Let us denote by $C$ the event that for all three rolls the number which comes up is 1 through 5, and by $D$ the event that the number 6 comes up at least once. When the probability $P(C)$ is compared with the probability $P(D)$, we see that $\mathbf { Y }$. (For $\mathbf { Y }$, choose the correct answer from among choices (0) $\sim$ (4) below.) (0) $P ( C )$ is less than $P ( D )$ and $P ( D )$ is not less than twice $P ( C )$.
(1) $P ( C )$ is less than $P ( D )$ and $P ( D )$ is less than twice $P ( C )$.
(2) $P ( C )$ and $P ( D )$ are the same.
(3) $P ( C )$ is greater than $P ( D )$ and $P ( C )$ is not less than twice $P ( D )$.
(4) $P ( C )$ is greater than $P ( D )$ and $P ( C )$ is less than twice $P ( D )$.