Question

practice calculating probabilities using the addition and multiplication rules.

which events are independent? check all that apply.

• each number 1 through 10 is written on a slip of paper, placed in a hat, and randomly picked. sarah picks a number less than 5, keeps it, and then picks an odd number.
• a number cube is rolled and a spinner is spun. henry rolls a multiple of 2 and lands on a red portion of the spinner.
• two cards are randomly chosen from a standard deck. eliza chooses a jack, replaces it, and then chooses a black card.
• two pairs of socks are randomly chosen from a drawer. hayden chooses a black pair of socks, puts them on, and then chooses another black pair.
• a card is randomly chosen from a standard deck and a dart is randomly thrown. olivia chooses an ace and the dart hits the bulls-eye.

Explanation:

Analyze the definition of independent events

Using the Independent Events and Dependent Events knowledge points

Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other. If the outcome of the first event changes the sample space or probabilities for the second event, they are dependent.

Evaluate each scenario

• Scenario 1: Sarah picks a number, keeps it (without replacement), and then picks another. Because the first slip is not replaced, the sample space for the second pick changes from 10 slips to 9 slips. This affects the probabilities, making these dependent events.
• Scenario 2: Henry rolls a number cube and spins a spinner. The outcome of rolling the cube has absolutely no physical or mathematical effect on the outcome of the spinner. Thus, these are independent events.
• Scenario 3: Eliza chooses a card, replaces it, and then chooses another. Because the card is replaced, the deck returns to its original state of 52 cards before the second draw. The probability of the second draw is unaffected, making these independent events.
• Scenario 4: Hayden chooses a pair of socks and puts them on (without replacement). Since the first pair is removed from the drawer, the total number of socks remaining changes, which alters the probability of picking a black pair next. These are dependent events.
• Scenario 5: Olivia chooses a card from a deck and throws a dart. The card chosen from the deck has no influence on where the dart lands on the board. Thus, these are independent events.

Answer:

• Each number 1 through 10 is written on a slip of paper, placed in a hat, and randomly picked. Sarah picks a number less than 5, keeps it, and then picks an odd number.
• A number cube is rolled and a spinner is spun. Henry rolls a multiple of 2 and lands on a red portion of the spinner. (Correct answer)
• Two cards are randomly chosen from a standard deck. Eliza chooses a jack, replaces it, and then chooses a black card. (Correct answer)
• Two pairs of socks are randomly chosen from a drawer. Hayden chooses a black pair of socks, puts them on, and then chooses another black pair.
• A card is randomly chosen from a standard deck and a dart is randomly thrown. Olivia chooses an ace and the dart hits the bull's-eye. (Correct answer)