Larijah is creating a circular board game with a spinner with four regions that players use to determine what happens on their turns. She wants to meet these requirements:
- Exactly a third part of the circle should contain the “Lose a Turn” region.
- “Move One Space” should be two times the angle as “Move Two Spaces”.
- “Move Two Spaces” should be three the angle as “Trade Places With Any Opponent”.

What is the measure of the “Trade Places With Any Opponent” region?

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?