Question

put the events in order of increasing probability.
you flip a nickel and it lands on heads.
you roll a 6 - sided die and it lands on 1, 3, 4, or 6.
you roll a 6 - sided die and it lands on a multiple of 2.
you pull a red ball from a bag of all red balls.
on a spinner with just 6, 8, and 10, you spin a 4.

Explanation:

Step1: Calculate probability for each event

- Flip nickel (heads): Possible outcomes = 2 (H/T), favorable = 1. Probability = $\frac{1}{2}$ = 0.5.
- Roll die (1,3,4,6): Favorable outcomes = 4, total = 6. Probability = $\frac{4}{6}$ ≈ 0.6667.
- Roll die (multiple of 2): Multiples of 2 on die: 2,4,6 (3 outcomes). Probability = $\frac{3}{6}$ = 0.5? Wait, no—wait, multiples of 2 on 1 - 6: 2,4,6 (3), so $\frac{3}{6}$ = 0.5? Wait, no, earlier event: "roll die and it lands on 1,3,4, or 6"—that's 4 outcomes (1,3,4,6), so $\frac{4}{6}$ ≈ 0.6667. Then "roll die and lands on multiple of 2": 2,4,6 (3), so $\frac{3}{6}$ = 0.5. Then "pull red ball from all red balls": Probability = 1 (certain). "Spin a 4 on spinner with 6,8,10": Possible outcomes = 3 (6,8,10), favorable = 0 (no 4). Probability = 0.

Wait, let's re - list each event:

1. Event 1: Flip nickel, lands on heads. Outcomes: H, T. P = $\frac{1}{2}$ = 0.5.
2. Event 2: Roll 6 - sided die, lands on 1,3,4, or 6. Favorable: 4 (1,3,4,6). Total: 6. P = $\frac{4}{6}$ = $\frac{2}{3}$ ≈ 0.6667.
3. Event 3: Roll 6 - sided die, lands on multiple of 2. Multiples of 2: 2,4,6 (3 outcomes). P = $\frac{3}{6}$ = 0.5.
4. Event 4: Pull red ball from bag of all red balls. All balls are red, so P = 1.
5. Event 5: Spin a 4 on spinner with 6,8,10. Spinner has 6,8,10—no 4. So favorable = 0. P = 0.

Wait, maybe I misread the "roll die and lands on multiple of 2"—wait, the die is 6 - sided (1 - 6). Multiples of 2: 2,4,6. So 3 outcomes. So P = 3/6 = 0.5.

Now, let's order by increasing probability (from lowest to highest):

- Event 5 (spin 4): P = 0.
- Then Event 1 (flip heads) and Event 3 (roll multiple of 2): both P = 0.5? Wait, no—wait, Event 1: P = 0.5, Event 3: P = 0.5. Then Event 2: P = 2/3 ≈ 0.6667. Then Event 4: P = 1.

Wait, but maybe the "roll die and lands on multiple of 2"—wait, maybe the die has numbers 1 - 6. Let's re - check each event:

1. "You flip a nickel and it lands on heads." P = 1/2 = 0.5.
2. "You roll a 6 - sided die and it lands on 1, 3, 4, or 6." Number of favorable: 4 (1,3,4,6). Total: 6. P = 4/6 = 2/3 ≈ 0.6667.
3. "You roll a 6 - sided die and it lands on a multiple of 2." Multiples of 2: 2,4,6. So 3. P = 3/6 = 0.5.
4. "You pull a red ball from a bag of all red balls." P = 1 (certain).
5. "On a spinner with just 6, 8, and 10, you spin a 4." Spinner has 3 sections (6,8,10). No 4. So P = 0.

Now, ordering from least to greatest probability:

- P = 0: "On a spinner with just 6, 8, and 10, you spin a 4."
- P = 0.5: "You flip a nickel and it lands on heads." and "You roll a 6 - sided die and it lands on a multiple of 2." (Wait, but maybe the "roll die and lands on multiple of 2"—wait, maybe I made a mistake. Wait, 2,4,6 are multiples of 2, so 3 out of 6. And flipping a nickel: 1 out of 2. Both 0.5. Then "You roll a 6 - sided die and it lands on 1, 3, 4, or 6": 4 out of 6 ≈ 0.6667. Then "You pull a red ball from a bag of all red balls": 1.

But the problem says "put the events in order of increasing probability". So the order (from lowest to highest) is:

1. On a spinner with just 6, 8, and 10, you spin a 4. (P = 0)
2. You flip a nickel and it lands on heads. (P = 0.5) or You roll a 6 - sided die and it lands on a multiple of 2. (P = 0.5)
3. You roll a 6 - sided die and it lands on 1, 3, 4, or 6. (P = 2/3 ≈ 0.6667)
4. You pull a red ball from a bag of all red balls. (P = 1)

Wait, but maybe the "roll die and lands on multiple of 2"—wait, maybe the die is 1 - 6, so multiples of 2 are 2,4,6 (3 numbers), so P = 3/6 = 0.5. Flipping a nickel: P = 1/2 = 0.5. So these two have the same probability. Then the next is the die with 4 favorable outcomes (1,3,4,6) with P = 4/6 = 2/3, then the red ball with P = 1. And the spinner with 4 has P = 0.

So the order from least to greatest probability is:

- On a spinner with just 6, 8, and 10, you spin a 4. (P = 0)
- You flip a nickel and it lands on heads. (P = 0.5)
- You roll a 6 - sided die and it lands on a multiple of 2. (P = 0.5)
- You roll a 6 - sided die and it lands on 1, 3, 4, or 6. (P = 2/3)
- You pull a red ball from a bag of all red balls. (P = 1)

(Note: The two events with P = 0.5 can be in either order relative to each other as they have the same probability.)

Answer:

1. On a spinner with just 6, 8, and 10, you spin a 4.
2. You flip a nickel and it lands on heads.
3. You roll a 6 - sided die and it lands on a multiple of 2.
4. You roll a 6 - sided die and it lands on 1, 3, 4, or 6.
5. You pull a red ball from a bag of all red balls.

(Or swap steps 2 and 3 as they have equal probability.)