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Chapter 3: Problem 19

Let an unbiased die be cast at random seven independent times. Compute the conditional probability that each side appears at least once given that side 1 appears exactly twice.

Short Answer

Expert verified
The conditional probability that in the course of 7 throws of the die, each side appears at least once given that side 1 appears exactly twice, is the ratio of \(\binom{7}{2}\) to 6.

Step by step solution

01

Compute the Total Ways for Side 1 to Appear Exactly Twice

This can be calculated using combinations because order doesn't matter when rolling the die. We need to calculate \(\binom{7}{2}\) which means choosing 2 successes (rolling a 1) from 7 trials.
02

Compute the Total Ways for All Sides to Appear at Least Once

Let's calculate the ways to throw the die 7 times so all faces appear at least once. First, let's allocate one appearance for each face of the die, so 1 throw for each side, which leaves us with one remainder throw. The ways for this remainder throw to give us any of the 6 faces is 6.
03

Compute the Conditional Probability

The conditional probability is the ratio of the two quantities computed in the previous steps. The main condition here being side 1 appearing exactly twice, which is what we calculated in Step 1. The event here is that each side appears at least once on throwing the die 7 times, which is what we calculated in Step 2.

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Most popular questions from this chapter

. Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote \(n\) mutually independent random variables with the moment-generating functions \(M_{1}(t), M_{2}(t), \ldots, M_{n}(t)\), respectively. (a) Show that \(Y=k_{1} X_{1}+k_{2} X_{2}+\cdots+k_{n} X_{n}\), where \(k_{1}, k_{2}, \ldots, k_{n}\) are real constants, has the mgf \(M(t)=\prod_{1}^{n} M_{i}\left(k_{i} t\right)\).

Let \(X\) and \(Y\) be independent random variables, each with a distribution that is \(N(0,1) .\) Let \(Z=X+Y .\) Find the integral that represents the cdf \(G(z)=\) \(P(X+Y \leq z)\) of \(Z .\) Determine the pdf of \(Z\). Hint: We have that \(G(z)=\int_{-\infty}^{\infty} H(x, z) d x\), where $$ H(x, z)=\int_{-\infty}^{z-x} \frac{1}{2 \pi} \exp \left[-\left(x^{2}+y^{2}\right) / 2\right] d y $$ Find \(G^{\prime}(z)\) by evaluating \(\int_{-\infty}^{\infty}[\partial H(x, z) / \partial z] d x\).

Show that the moment generating function of the negative binomial distribution is \(M(t)=p^{r}\left[1-(1-p) e^{t}\right]^{-r}\). Find the mean and the variance of this distribution. Hint: In the summation representing \(M(t)\), make use of the MacLaurin's series for \((1-w)^{-r}\)

Compute \(P\left(X_{1}+2 X_{2}-2 X_{3}>7\right)\), if \(X_{1}, X_{2}, X_{3}\) are iid with common distribution \(N(1,4)\).

If \(M\left(t_{1}, t_{2}\right)\) is the mgf of a bivariate normal distribution, compute the covariance by using the formula $$ \frac{\partial^{2} M(0,0)}{\partial t_{1} \partial t_{2}}-\frac{\partial M(0,0)}{\partial t_{1}} \frac{\partial M(0,0)}{\partial t_{2}} $$ Now let \(\psi\left(t_{1}, t_{2}\right)=\log M\left(t_{1}, t_{2}\right) .\) Show that \(\partial^{2} \psi(0,0) / \partial t_{1} \partial t_{2}\) gives this covariance directly.
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