1) What is the probability of tossing 6 coins and getting at least two tails?

2) Suppose you flip a penny, a nickel,a dime and a quarter. What is the probability of getting heads on the penny and nickel and dime and tails on the quarter?

Please show steps for the solutions.
Thank you.

p tails in 1 toss = .5

binomial distribution
p = .5
number of trials , n , = 6
number of successes = 2,3,4,5,6
number of failures = 0,1
so easier to work it backwards
what is the probability of 0 or 1 heads
p is still .5
p(0 heads) = C(6,0) (.5^0)(.5)^6 = .0156
p(1 heads) = C(6,1)(.5^1)(.5)^5 = 3*.03125 = .09375
sum = about .11
1 - .11 = .89

Suppose you flip a penny, a nickel,a dime and a quarter. What is the probability of getting heads on the penny and nickel and dime and tails on the quarter?

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independent experiments
1/2*1/2*1/2*1/2

a dime rolled 5 times find the probability of getting 4 exactly 5 times

To calculate the probabilities of the given scenarios, we need to use the concept of probability and combinations.

1) Probability of tossing 6 coins and getting at least two tails:

Step 1: Determine the total number of possible outcomes.
When tossing a coin, there are 2 possible outcomes: heads or tails. Therefore, when tossing 6 coins, the total number of possible outcomes is 2^6 = 64.

Step 2: Determine the number of favorable outcomes.
We need to find the number of outcomes where at least two tails occur. To do this, we can use the concept of combinations. We can have 6 tails, 5 tails, 4 tails, 3 tails, or 2 tails.

To calculate the number of combinations, we use the formula C(n, r) = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items selected.

Number of outcomes with 6 tails: C(6, 6) = 6! / (6! * (6-6)!) = 1
Number of outcomes with 5 tails: C(6, 5) = 6! / (5! * (6-5)!) = 6
Number of outcomes with 4 tails: C(6, 4) = 6! / (4! * (6-4)!) = 15
Number of outcomes with 3 tails: C(6, 3) = 6! / (3! * (6-3)!) = 20
Number of outcomes with 2 tails: C(6, 2) = 6! / (2! * (6-2)!) = 15

Step 3: Calculate the probability.
The probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Number of favorable outcomes = 1 + 6 + 15 + 20 + 15 = 57
Probability = Number of favorable outcomes / Total number of possible outcomes = 57 / 64 ≈ 0.891

Therefore, the probability of tossing 6 coins and getting at least two tails is approximately 0.891, or 89.1%.

2) Probability of getting heads on the penny, nickel, and dime, and tails on the quarter:

Step 1: Determine the total number of possible outcomes.
When flipping a single coin, there are two possible outcomes: heads or tails. Therefore, when flipping a penny, a nickel, a dime, and a quarter, the total number of possible outcomes is 2 * 2 * 2 * 2 = 16.

Step 2: Determine the number of favorable outcomes.
We are looking for the outcome where the penny, nickel, and dime land on heads, and the quarter lands on tails. Since each coin flip is independent, the number of favorable outcomes is simply 1.

Step 3: Calculate the probability.
The probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Number of favorable outcomes = 1
Probability = Number of favorable outcomes / Total number of possible outcomes = 1 / 16 ≈ 0.0625

Therefore, the probability of getting heads on the penny, nickel, and dime, and tails on the quarter is approximately 0.0625, or 6.25%.