Probability Questions & Answers

Determine whether the following statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If an experiment is repeated more and more times, the theoretical probability of an event tends to get closer to the empirical probability of that event. A. The statement is true because the theoretical probability of an event tends to get closer to the empirical probability of that event when an experiment is repeated more and more times. B. The statement is true because if ahsexperiment is repeated more and more times, the proportion of outcomes favorable to any particular even tends to get closer to the theoretical probability of that event. C. The statement is false. The true statement is "If an experiment is repeated more and more times, the empirical probability of that event tends to get closer to 1 ." D. The statement is false. The true statement is "If an experiment is repeated more and more times, the empirical probability of an event tends to get closer to the theoretical probability of that event."

11. 50 people were asked if they speak French or German or Spanish. Of these people, 31 speak French 2 speak French, German and Spanish 4 speak French and Spanish but not German 7 speak German and Spanish 8 do not speak any of the languages All 10 people who speak German speak at least one other language. Two of the 50 people are chosen at random. Work out the probability that they both only speak Spanish.

According to a study, \( 76 \% \) of \( \mathrm{K}-12 \) schools or districts in a country use digital content such as ebooks, audio books, and digital textbooks. Of these \( 76 \%, 4 \) out of 10 use digital content as part of their curriculum. Find the probability that a randomly selected school or district uses digital content and uses it as part of their curriculum. The probability that a randomly selected school or district uses digital content and uses it as part of their curriculum is (Round to three decimal places as needed.)

In a recent year, about \( 36 \% \) of all infants born in a country were conceived through in-vitro fertilization (IVF). Of the IVF deliveries, about twenty-six percent resulted in multiple births. (a) Find the probability that a randomly selected infant was conceived through IVF and was part of a multiple birth. (b) Find the probability that a randomly selected infant conceived through IVF was not part of a multiple birth. (c) Would it be unusual for a randomly selected infant to have been conceived through IVF and to have been part of a multiple birth? Explain. (a) The probability that a randomly selected infant was conceived through IVF and was part of a multiple birth is (Round to the nearest thousandth as needed.)

In a recent year, about \( 34 \% \) of all infants born in a country were conceived through in-vitro fertilization (IVF). Of the IVF deliveries, about twenty-two percent resulted in miltiple births. (a) Find the probability that a randomly selected infant was conceived through IVF and was part of a multiple birth. (b) Find the probability that a randomly selected infant conceived through IVF was not part of a multiple birth. (c) Would it be unusual for a randomly selected infant to.have been conceived through IVF and to have been part of a multiple birth? Explain. (a) The probability that a randomly selected infant was conceived through IVF and was part of a multiple birth is (Round to the nearest thousandth as needed.)

The life expectancy (in years) of a certain brand of clock radio is a continuous random variable with the probability density function below. \( f(x)=\left\{\begin{array}{ll}2 /(x+2)^{2} & \text { if } x \geq 0 \\ 0 & \text { otherwise }\end{array}\right. \) (A) Find the probability that a randomly selected clock lasts at most 7 years. (B) Find the probability that a randomly selected clock radio lasts from 7 to 11 years. (C) Graph \( y=f(x) \) for \( [0,11] \) and show the shaded region for part (A). (A) What is the probability that a clock will last no more than 7 years? (Type a decimal rounded to three decimal places as needed.)

There are 20 counters in a bag. 2 of the counters are white. 1 of the counters is pink. 4 of the counters are black. The rest of the counters are purple. Carter takes a counter at random from the bag. Show that the probability that the counter is white or purple is \( \frac{3}{4} \)

A manufacturer guarantees a product for 1 year. The lifespan of the product after it is sold is given by the probability density function below, where \( t \) is time in months. \( f(t)=\left\{\begin{array}{ll}0.012 e^{-0.01 t} & \text { if } t \geq 0 \\ 0 & \text { otherwise }\end{array}\right. \) What is the probability that a buyer chosen at random will have a product failure (A) During the warranty? (B) During the second year after purchase? (A) What is the probability that the product will fail within one year? 0.136 (Round to three decimal places as needed.) (B) What is the probability that the product will fail during the second year after purchase? (Round to three decimal places as needed.)

A manufacturer guarantees a product for 1 year. The lifespan of the product after it is sold is given by the probability density function below, where \( t \) is time in months. \( f(t)=\left\{\begin{array}{ll}0.012 e^{-0.01 t} & \text { if } t \geq 0 \\ 0 & \text { otherwise }\end{array}\right. \) What is the probability that a buyer chosen at random will have a product failure \( \begin{array}{l}\text { (A) During the warranty? } \\ \text { (B) During the second year after purchase? }\end{array} \) (A) What is the probability that the product will fail within one year? (Round to three decimal places as needed.)

The life expectancy (in years) of a certain brand of clock radio is a continuous random variable with the probability density function below. \( f(x)=\left\{\begin{array}{ll}2 /(x+2)^{2} & \text { if } x \geq 0 \\ 0 & \text { otherwise }\end{array}\right. \) \( \begin{array}{l}\text { (A) Find the probability that a randomly selected clock lasts at most } 5 \text { years. } \\ \text { (B) Find the probability that a randomly selected clock radio lasts from } 5 \text { to } 11 \text { years. } \\ \text { (C) Graph } y=f(x) \text { for }[0,11] \text { and show the shaded region for part (A). } \\ \text { (A) What is the probability that a clock will last no more than } 5 \text { years? } \\ \text { (Type a decimal rounded to three decimal places as needed.) }\end{array} \).