Question

alice likes to play a trivia game on her phone. she is able to answer the question correctly 75% of the time. how likely is it that she will answer 5 questions correctly in a row? which simulation could be used to fairly represent the situation? create a deck of 75 cards, each labeled with a different number from 1 to 75. pick a card from the deck, then return it to the deck, 5 times. each time a number less than or equal to 50 appears, it represents alice answering a question correctly. put 3 red marbles and 1 blue marble in a bag. draw a marble from the bag, then return it to the bag, 5 times. each time a red marble appears, it represents alice answering a question correctly. create a spinner with 4 equal sections. color 3 sections green and 1 section red. spin the spinner 75 times. each time the spinner lands on a green section, it represents alice answering a question correctly.

Explanation:

Step1: Identify the probability of answering one question correctly

The probability of answering a question correctly is $p = 0.75=\frac{3}{4}$.

Step2: Use the multiplication - rule for independent events

Since the events of answering each question correctly are independent, the probability of answering 5 questions correctly in a row is $P=(0.75)^5$.
$P = (\frac{3}{4})^5=\frac{3^5}{4^5}=\frac{243}{1024}\approx0.237$.

Step3: Analyze the simulations

• For the card - picking simulation: The probability of picking a number less than or equal to 50 out of 75 cards is $\frac{50}{75}=\frac{2}{3}

eq0.75$.

• For the marble - drawing simulation: There are 3 red marbles and 1 blue marble. The probability of drawing a red marble is $\frac{3}{3 + 1}=0.75$, which is the same as the probability of Alice answering a question correctly.
• For the spinner simulation: The probability of landing on a green section is $\frac{3}{4}=0.75$, but it is spun 75 times instead of 5 times as required in the situation of answering 5 questions.

Answer:

1. The probability that she will answer 5 questions correctly in a row is $\frac{243}{1024}\approx0.237$.
2. Put 3 red marbles and 1 blue marble in a bag. Draw a marble from the bag, then return it to the bag, 5 times. Each time a red marble appears, it represents Alice answering a question correctly.