A penny, a nickel, and a dime are flipped at the same time. Each coin can come out either heads (H) or tails (T).
a. What name is given to the act of flipping the coins?
b. There are eight elements in the sample space (i.e., HHH,THT, etc.). List all eight outcomes.
c. How many outcomes are there in the event, "Exactly two of the coins are heads"
d. Calculate P(HHT).
e. Calculate P(exactly two heads).
f. Calculate P (at least two heads).
g. Calculate P (penny and nickel are tails).
h. Calculate P (penny or nickel are tails).
i. Calculate P (none is tails).
j. Calculate P (zero, one, two or three heads).
k. Calculate P (four heads).
p n d
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T
C two heads ---> 3
d P(HHT) = 1/8
e 3/8
f 4/8 = 1/2
g 2/8 = 1/4
h 6/8 = 3/4
i 1/8
j 1
k zero, not in set
a. The name given to the act of flipping the coins is "Coin flipping extravaganza!"
b. The eight outcomes are:
1. HHH
2. HHT
3. HTH
4. HTT
5. THH
6. THT
7. TTH
8. TTT
c. There are three outcomes in the event "Exactly two of the coins are heads": HHT, HTH, and THH.
d. P(HHT) is 1/8 or 0.125 (or as I like to call it, the chance of a happy heads transformation).
e. P(exactly two heads) is 3/8 or 0.375. That's three out of the eight outcomes or "Heads-Up Magic!"
f. P(at least two heads) is 4/8 or 0.5. That's four out of the eight outcomes or "Heads up, high chance!"
g. P(penny and nickel are tails) is 2/8 or 0.25. That's two out of the eight outcomes or "Tales of the Tails!"
h. P(penny or nickel are tails) is 7/8 or 0.875. That's seven out of the eight outcomes or "A tale of many tails!"
i. P(none is tails) is 1/8 or 0.125. That's one out of the eight outcomes or "A magical tale of all heads!"
j. P(zero, one, two, or three heads) is 1 because it covers all the outcomes. You've got it all covered, my friend!
k. P(four heads) is 0/8 or 0. That's zero out of the eight outcomes or "No heads, all heart!"
a. The act of flipping the coins is called a coin toss.
b. The eight outcomes in the sample space are:
1. HHH
2. HHT
3. HTH
4. HTT
5. THH
6. THT
7. TTH
8. TTT
c. There are three outcomes in the event "Exactly two of the coins are heads": HHT, HTH, and THH.
d. To calculate P(HHT), we divide the number of favorable outcomes (1) by the total number of outcomes (8). Therefore, P(HHT) = 1/8.
e. To calculate P(exactly two heads), we add up the probabilities of the three outcomes in the event "Exactly two of the coins are heads" and divide by the total number of outcomes. Therefore, P(exactly two heads) = 3/8.
f. To calculate P(at least two heads), we need to consider the outcomes with two or more heads. These are HHH, HHT, HTH, and THH. Therefore, P(at least two heads) = 4/8 = 1/2.
g. To calculate P(penny and nickel are tails), we count the outcomes where both the penny and nickel are tails. There is only one outcome that satisfies this condition: TTT. Therefore, P(penny and nickel are tails) = 1/8.
h. To calculate P(penny or nickel are tails), we count the outcomes where at least one of the penny or nickel is tails. These outcomes are TTT, TTH, THT, HTT, and THH. Therefore, P(penny or nickel are tails) = 5/8.
i. To calculate P(none is tails), we count the outcomes where all three coins are heads. There is only one outcome that satisfies this condition: HHH. Therefore, P(none is tails) = 1/8.
j. To calculate P(zero, one, two or three heads), we add up the probabilities of all outcomes except when all four coins are heads (HHH). Therefore, P(zero, one, two or three heads) = 7/8.
k. To calculate P(four heads), we need to consider the outcome where all four coins are heads (HHHH). There is no such outcome in this scenario, as we only have three coins. Therefore, P(four heads) = 0/8 = 0.
a. The name given to the act of flipping the coins is called a coin toss.
b. The eight outcomes in the sample space are:
1. HHH (all coins are heads)
2. HHT (two heads, one tail)
3. HTH (one head, two tails)
4. HTT (one head, two tails)
5. THH (two heads, one tail)
6. THT (one head, two tails)
7. TTH (one head, two tails)
8. TTT (all coins are tails)
c. To determine the number of outcomes in the event "Exactly two of the coins are heads," we can identify the specific outcomes that satisfy this condition. In this case, the outcomes are HHT, HTH, and THH. Therefore, there are three outcomes.
d. To calculate P(HHT), we need to determine the probability of getting the specific outcome HHT. Since there are eight total outcomes in the sample space, and only one of them is HHT, the probability is 1/8.
e. To calculate the probability of "exactly two heads," we need to determine the probability of getting HHT, HTH, or THH. Since each of these outcomes has a probability of 1/8, and there are three of them, the probability of getting exactly two heads is 3/8.
f. To calculate the probability of "at least two heads," we need to determine the probabilities of getting exactly two heads or all three heads. The probability of getting exactly two heads is 3/8, and the probability of getting all three heads is 1/8. Adding these probabilities gives us 4/8, which simplifies to 1/2.
g. To calculate the probability that the penny and nickel are tails, we need to consider the outcome TTT. Since there is only one outcome with this condition, and there are eight total outcomes, the probability is 1/8.
h. To calculate the probability that either the penny or nickel is tails, we need to consider the outcomes that have at least one tail. These outcomes are HTT, THT, TTH, and TTT. Since there are four outcomes out of eight total, the probability is 4/8, which simplifies to 1/2.
i. To calculate the probability that none of the coins is tails, we need to consider the outcome HHH. Since there is only one outcome with this condition, and there are eight total outcomes, the probability is 1/8.
j. To calculate the probability of "zero, one, two, or three heads," we need to consider all outcomes except the one with four heads (HHH). Since there are seven outcomes out of eight total, the probability is 7/8.
k. To calculate the probability of "four heads," we need to determine the probability of the specific outcome HHH. Since there is only one outcome with this condition, and there are eight total outcomes, the probability is 1/8.