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Answer: 0.53; p = 0, q = 0.12; r = 0.1, t = 0.2; independent: 0.48 × 0.25 = 0.12คำตอบ: 0.53; p = 0, q = 0.12; r = 0.1, t = 0.2; independent: 0.48 × 0.25 = 0.12
1(a) Exactly one magazine1(a) อ่านแค่เล่มเดียว
A college prints three magazines, on green issues, equality and sports. One student is picked at random. $G$, $E$ and $S$ are the events that the student reads the green issues, the equality and the sports magazine.
The Venn diagram gives the probability of each part: $G$ only 0.08, $G$ and $E$ only 0.05, $E$ only 0.09 and $S$ only 0.36. The letters $p$, $q$, $r$ and $t$ are the other four parts: $p$ is all three magazines, $q$ is $G$ and $S$ only, $r$ is $E$ and $S$ only, and $t$ is none of them.
(a) What proportion of the students read exactly one magazine? [1]
1(b) Find p and q1(b) หา p กับ q
A college prints three magazines, on green issues, equality and sports. One student is picked at random. $G$, $E$ and $S$ are the events that the student reads the green issues, the equality and the sports magazine.
The Venn diagram gives the probability of each part: $G$ only 0.08, $G$ and $E$ only 0.05, $E$ only 0.09 and $S$ only 0.36. The letters $p$, $q$, $r$ and $t$ are the other four parts: $p$ is all three magazines, $q$ is $G$ and $S$ only, $r$ is $E$ and $S$ only, and $t$ is none of them.
(b) Nobody reads all three magazines, and $\mathrm{P}(G)=0.25$. Find (i) $p$ and (ii) $q$. [3]
1(c) Find r and t1(c) หา r กับ t
A college prints three magazines, on green issues, equality and sports. One student is picked at random. $G$, $E$ and $S$ are the events that the student reads the green issues, the equality and the sports magazine.
The Venn diagram gives the probability of each part: $G$ only 0.08, $G$ and $E$ only 0.05, $E$ only 0.09 and $S$ only 0.36. The letters $p$, $q$, $r$ and $t$ are the other four parts: $p$ is all three magazines, $q$ is $G$ and $S$ only, $r$ is $E$ and $S$ only, and $t$ is none of them.
(c) Use $p=0$ and $q=0.12$. Given that $\mathrm{P}(S\mid E)=\dfrac{5}{12}$, find (i) $r$ and (ii) $t$. [4]
1(d) Test independence1(d) ทดสอบความเป็นอิสระ
A college prints three magazines, on green issues, equality and sports. One student is picked at random. $G$, $E$ and $S$ are the events that the student reads the green issues, the equality and the sports magazine.
The Venn diagram gives the probability of each part: $G$ only 0.08, $G$ and $E$ only 0.05, $E$ only 0.09 and $S$ only 0.36. The letters $p$, $q$, $r$ and $t$ are the other four parts: $p$ is all three magazines, $q$ is $G$ and $S$ only, $r$ is $E$ and $S$ only, and $t$ is none of them.
(d) Using $q=0.12$, decide whether the events $S\cap E'$ and $G$ are independent. Show every step. [3]
Answer: 0.38; p = 0.2, q = 0.07; 13/40 (= 0.325)คำตอบ: 0.38; p = 0.2, q = 0.07; 13/40 (= 0.325)
2(a) Find P(A)2(a) หา P(A)
The Venn diagram shows the probabilities of three events $A$, $B$ and $C$. The parts are: $A$ only 0.13, $A$ and $B$ only 0.25, $B$ only 0.05, and $B$ and $C$ only 0.3. The letters $p$ and $q$ are probabilities too: $p$ is $C$ only and $q$ is outside all three circles. The circles for $A$ and $C$ do not overlap.
(a) Find $\mathrm{P}(A)$. [1]
2(b) Find p and q2(b) หา p กับ q
The Venn diagram shows the probabilities of three events $A$, $B$ and $C$. The parts are: $A$ only 0.13, $A$ and $B$ only 0.25, $B$ only 0.05, and $B$ and $C$ only 0.3. The letters $p$ and $q$ are probabilities too: $p$ is $C$ only and $q$ is outside all three circles. The circles for $A$ and $C$ do not overlap.
(b) Events $B$ and $C$ are independent. Work out $p$ and $q$. [3]
2(c) Find P(A | B′)2(c) หา P(A | B′)
The Venn diagram shows the probabilities of three events $A$, $B$ and $C$. The parts are: $A$ only 0.13, $A$ and $B$ only 0.25, $B$ only 0.05, and $B$ and $C$ only 0.3. The letters $p$ and $q$ are probabilities too: $p$ is $C$ only and $q$ is outside all three circles. The circles for $A$ and $C$ do not overlap.
(c) Using $p=0.2$ and $q=0.07$, find $\mathrm{P}(A\mid B')$. [2]
Answer: a = 2, b = 3, c = 6; 0.381คำตอบ: a = 2, b = 3, c = 6; 0.381
3(a) Find a, b, c3(a) หา a, b, c
The discrete random variable $X$ takes the values $a$, $b$ and $c$, which are different integers with $a\lt b\lt c$. Their probabilities are $\mathrm{P}(X=a)=\log_{36}a$, $\mathrm{P}(X=b)=\log_{36}b$ and $\mathrm{P}(X=c)=\log_{36}c$, and every one of them is greater than 0.
(a) Work out $a$, $b$ and $c$, showing every step. [5]
3(b) P(X₁ = X₂)3(b) P(X₁ = X₂)
The discrete random variable $X$ takes the values $a$, $b$ and $c$, which are different integers with $a\lt b\lt c$. Their probabilities are $\mathrm{P}(X=a)=\log_{36}a$, $\mathrm{P}(X=b)=\log_{36}b$ and $\mathrm{P}(X=c)=\log_{36}c$, and every one of them is greater than 0.
(b) $X_1$ and $X_2$ are independent copies of $X$ (they have the same distribution). Using the values from (a), find $\mathrm{P}(X_1=X_2)$. [2]
Answer: c = 8b/5; a = 2/25, b = 1/5, c = 8/25, d = 2/5; k = 30, so P(S | X = 20) = 1.5, which is more than 1คำตอบ: c = 8b/5; a = 2/25, b = 1/5, c = 8/25, d = 2/5; k = 30, so P(S | X = 20) = 1.5, which is more than 1
4(a) Show c = 8b/54(a) แสดงว่า c = 8b/5
Tisam plays a game with a ball, a cup and a spinner. The spinner gives a score $X$ of 20, 50, 80 or 100, with probabilities $a$, $b$, $c$ and $d$ in that order. If it lands on $x$, she stands $x$ cm from the cup and throws the ball at it. $S$ is the event that the ball lands in the cup.
Her model: $\mathrm{P}(S\mid\{X=x\})=\dfrac{k}{x}$, where $k$ is a constant, and $\mathrm{P}(S\cap\{X=x\})$ has the same value for every $x$.
(a) Using her model, show that $c=\dfrac{8b}{5}$. [2]
4(b) Distribution of X4(b) การแจกแจงของ X
Tisam plays a game with a ball, a cup and a spinner. The spinner gives a score $X$ of 20, 50, 80 or 100, with probabilities $a$, $b$, $c$ and $d$ in that order. If it lands on $x$, she stands $x$ cm from the cup and throws the ball at it. $S$ is the event that the ball lands in the cup.
Her model: $\mathrm{P}(S\mid\{X=x\})=\dfrac{k}{x}$, where $k$ is a constant, and $\mathrm{P}(S\cap\{X=x\})$ has the same value for every $x$.
(b) Using her model, find $a$, $b$, $c$ and $d$. [5]
4(c) Why the model fails4(c) ทำไมโมเดลใช้ไม่ได้
Tisam plays a game with a ball, a cup and a spinner. The spinner gives a score $X$ of 20, 50, 80 or 100, with probabilities $a$, $b$, $c$ and $d$ in that order. If it lands on $x$, she stands $x$ cm from the cup and throws the ball at it. $S$ is the event that the ball lands in the cup.
Her model: $\mathrm{P}(S\mid\{X=x\})=\dfrac{k}{x}$, where $k$ is a constant, and $\mathrm{P}(S\cap\{X=x\})$ has the same value for every $x$.
(c) Nav throws many times from 100 cm and the ball goes in the cup 30% of the time. Explain why Tisam's model cannot describe Nav for every score $X$. [1]
Answer: the chance of a hit is constant; the throws are independent; 0.0128; 0.0656; 0.02; 0.09; Peta: the chance of a hit is constant, Thomas: it increases with each throwคำตอบ: the chance of a hit is constant; the throws are independent; 0.0128; 0.0656; 0.02; 0.09; Peta: the chance of a hit is constant, Thomas: it increases with each throw
5(a) Two assumptions5(a) สมมติฐานสองข้อ
Children each throw a dart at a target many times. $H$ is the number of hits in a child's first 10 throws, and Peta models $H$ as $\mathrm{B}(10,\ 0.1)$. $F$ is the number of the throw on which a child first hits the target.
(a) Give two things Peta has to assume about the throws for $\mathrm{B}(10,\ 0.1)$ to fit. [2]
5(b) Find P(H ≥ 4)5(b) หา P(H ≥ 4)
Children each throw a dart at a target many times. $H$ is the number of hits in a child's first 10 throws, and Peta models $H$ as $\mathrm{B}(10,\ 0.1)$. $F$ is the number of the throw on which a child first hits the target.
(b) Using $H\sim\mathrm{B}(10,\ 0.1)$, find $\mathrm{P}(H\ge4)$. [1]
5(c) First hit on throw 55(c) โดนครั้งแรกที่ลูกที่ 5
Children each throw a dart at a target many times. $H$ is the number of hits in a child's first 10 throws, and Peta models $H$ as $\mathrm{B}(10,\ 0.1)$. $F$ is the number of the throw on which a child first hits the target.
(c) Keep Peta's assumptions (independent throws, a hit has chance 0.1 each time). Find $\mathrm{P}(F=5)$. [2]
5(d) Find α5(d) หา α
Children each throw a dart at a target many times. $H$ is the number of hits in a child's first 10 throws, and Peta models $H$ as $\mathrm{B}(10,\ 0.1)$. $F$ is the number of the throw on which a child first hits the target.
(d) Thomas assumes that no child needs more than 10 throws to hit the target for the first time. His model is $\mathrm{P}(F=n)=0.01+(n-1)\alpha$ for $n=1,\ 2,\ \dots,\ 10$, where $\alpha$ is a constant. Find $\alpha$. [4]
5(e)(f) Throw 5 · compare models5(e)(f) ลูกที่ 5 · เทียบสองโมเดล
Children each throw a dart at a target many times. $H$ is the number of hits in a child's first 10 throws, and Peta models $H$ as $\mathrm{B}(10,\ 0.1)$. $F$ is the number of the throw on which a child first hits the target.
(e)(f) Using Thomas' model with $\alpha=0.02$, find $\mathrm{P}(F=5)$. Then say how Peta's model and Thomas' model differ about the chance that a dart hits the target. [2]
Same method, new numbers. Write what you know first.วิธีเดิม ตัวเลขใหม่ · จดสิ่งที่รู้ก่อน
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