probability and tree diagrams\nuse a probability tree to help answer the following questions.\na regular die…

probability and tree diagrams\nuse a probability tree to help answer the following questions.\na regular die has 6 sides and a coin has 2 sides. you roll the die once then flip the coin once.\nwhat is the probability of:\n1) rolling a 5 and flipping a heads?\n2) rolling a 5 or flipping a heads?\n3) rolling an even number and flipping a tails?\n4) rolling an even number or flipping a tails?\na bag has 2 marbles labeled 4 and 7 and a spinner has 3 options. you randomly pull one marble then spin the spinner once.\nwhat is the probability of:\n5) drawing a number less than 8 or spinning a 4?\n6) drawing a number less than 8 and spinning a 4?\n7) drawing an 8 or spinning a 3?\n8) drawing an 8 and spinning a 3?

probability and tree diagrams\nuse a probability tree to help answer the following questions.\na regular die has 6 sides and a coin has 2 sides. you roll the die once then flip the coin once.\nwhat is the probability of:\n1) rolling a 5 and flipping a heads?\n2) rolling a 5 or flipping a heads?\n3) rolling an even number and flipping a tails?\n4) rolling an even number or flipping a tails?\na bag has 2 marbles labeled 4 and 7 and a spinner has 3 options. you randomly pull one marble then spin the spinner once.\nwhat is the probability of:\n5) drawing a number less than 8 or spinning a 4?\n6) drawing a number less than 8 and spinning a 4?\n7) drawing an 8 or spinning a 3?\n8) drawing an 8 and spinning a 3?

Answer

Explanation:

Step1: Recall probability formulas

For independent - events (A) and (B), (P(A\cap B)=P(A)\times P(B)) and (P(A\cup B)=P(A)+P(B)-P(A\cap B)).

Step2: Calculate probabilities for die - coin case

  1. The probability of rolling a 5 on a 6 - sided die is (P(\text{roll }5)=\frac{1}{6}), and the probability of flipping a head on a coin is (P(\text{head})=\frac{1}{2}).
    • For (P(\text{roll }5\cap\text{head})), since they are independent events, (P(\text{roll }5\cap\text{head})=\frac{1}{6}\times\frac{1}{2}=\frac{1}{12}).
  2. (P(\text{roll }5)=\frac{1}{6}), (P(\text{head})=\frac{1}{2}), and (P(\text{roll }5\cap\text{head})=\frac{1}{12}).
    • Using the formula (P(A\cup B)=P(A)+P(B)-P(A\cap B)), (P(\text{roll }5\cup\text{head})=\frac{1}{6}+\frac{1}{2}-\frac{1}{12}=\frac{2 + 6-1}{12}=\frac{7}{12}).
  3. The probability of rolling an even number on a 6 - sided die ((2,4,6)) is (P(\text{even})=\frac{3}{6}=\frac{1}{2}), and the probability of flipping a tail on a coin is (P(\text{tail})=\frac{1}{2}).
    • For (P(\text{even}\cap\text{tail})), since they are independent events, (P(\text{even}\cap\text{tail})=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}).
  4. (P(\text{even})=\frac{1}{2}), (P(\text{tail})=\frac{1}{2}), and (P(\text{even}\cap\text{tail})=\frac{1}{4}).
    • Using the formula (P(A\cup B)=P(A)+P(B)-P(A\cap B)), (P(\text{even}\cup\text{tail})=\frac{1}{2}+\frac{1}{2}-\frac{1}{4}=\frac{4 - 1}{4}=\frac{3}{4}).

Step3: Calculate probabilities for marble - spinner case

The probability of drawing a number less than 8 from the bag (both 4 and 7 are less than 8) is (P(\text{less than }8)=1), and the probability of spinning a 4 on a 3 - option spinner is (P(\text{spin }4)=0). 5. (P(\text{less than }8\cup\text{spin }4)=P(\text{less than }8)+P(\text{spin }4)-P(\text{less than }8\cap\text{spin }4)=1 + 0-0 = 1). 6. (P(\text{less than }8\cap\text{spin }4)=P(\text{less than }8)\times P(\text{spin }4)=1\times0 = 0). The probability of drawing an 8 from the bag is (P(\text{draw }8)=0), and the probability of spinning a 3 on a 3 - option spinner is (P(\text{spin }3)=\frac{1}{3}). 7. (P(\text{draw }8\cup\text{spin }3)=P(\text{draw }8)+P(\text{spin }3)-P(\text{draw }8\cap\text{spin }3)=0+\frac{1}{3}-0=\frac{1}{3}). 8. (P(\text{draw }8\cap\text{spin }3)=P(\text{draw }8)\times P(\text{spin }3)=0\times\frac{1}{3}=0).

Answer:

  1. (\frac{1}{12})
  2. (\frac{7}{12})
  3. (\frac{1}{4})
  4. (\frac{3}{4})
  5. (1)
  6. (0)
  7. (\frac{1}{3})
  8. (0)