This post covers Introduction to probability from Statistics for Engineers and Scientists by William Navidi.

Exercises

• A message consists of a string of bits ($0s$ and $1s$). Due to noise in the communications channel, each bit has probability $0.3$ of being reversed (i.e., a $1$ will be changed to a $0$ or a $0$ to a $1$). To improve the accuracy of the communication, each bit is sent five times, so, for example, $0$ is sent as $00000$. The receiver assigns the value $0$ if three or more of the bits are decoded as $0$, and $1$ if three or more of the bits are decoded as $1$. Assume that errors occur independently.
  • A $0$ is sent (as $00000$). What is the probability that the receiver assigns the correct value of 0?
  • Assume that each bit is sent n times, where n is an odd number, and that the receiver assigns the value decoded in the majority of the bits. What is the minimum value of n necessary so that the probability that the correct value is assigned is at least $0.90$?

Solution: (a) $0.8369$ (b) $9$

• Let$ X ∼ Bin(n, p)$, and let $Y = n − X$. Show that $Y ∼ Bin(n, 1 − p)$.

• Porcelain figurines are sold for $10 $ $ if flawless, and for $ 3 $ $ if there are minor cosmetic flaws. Of the figurines made by a certain company, $90%$ are flawless and $10%$ have minor cosmetic flaws. In a sample of $100$ figurines that are sold, let $Y$ be the revenue earned by selling them and let $X$ be the number of them that are flawless.

  a. Express $Y$ as a function of $X$.

  b. Find $\mu_Y$ .

  c. Find $\sigma_Y$ .

Solution: (a) $Y = 7X + 300$ (b) $930$ (c) $21$

• In a sample of $100$ newly manufactured automobile tires, $7$ are found to have minor flaws in the tread. If four newly manufactured tires are selected at random and installed on a car, Estimate the probability that exactly one of the four tires has a flaw, and find the uncertainty in the estimate.

Solution: $0.225 \pm 0.064$