Probabilities of Unfair Coin Flipping

In summary, the conversation is discussing the probability of getting exactly two heads and two tails when flipping an unfair coin four times in a row. The probability of getting heads is P(H)=0.40 and the probability of getting tails is P(T)=0.60. The formula for calculating P(2H) and P(2T) is incorrect, with the correct probabilities being 0.5248 for at least two heads and 0.8208 for at least two tails according to the binomial distribution. The speaker is seeking clarification on how to solve the problem and mentions a large tree diagram that may be incorrect.
  • #1
opticaltempest
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I am having a difficult time on the following problem:

An unfair coin is flipped four times in a row. What is the probability of getting exactly two heads and two tails. The order does not matter as long as there are two head and two tails in the flip.

The probability of getting heads is P(H)=0.40
The probability of getting tails is P(T)=0.60

I tried this:

P(2H) = 4C2 * 0.40^4 = 0.1536

P(2T) = 4C2 * 0.60^4 = 0.7776

Which has me stumped because the probability of getting at least two heads should be 0.5248 according to my large tree diagram. What exactly is my P(2H) formula calculating? The P(2T) seems to correctly calculate the probability of getting at least two tails, according to my tree diagram. Any hints on how to solve this problem?
 
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  • #2
What you are doing is wrong. Why are you taking [itex]4C2 P(H)^4[/itex] and [itex]4C2 P(T)^4[/itex]?

I don't know what your large tree diagram is. Your tree is right when it gives the probability of getting at least two heads is 0.5248. However, if your tree says the probability of getting at least two tails is 0.7776, it is wrong. The correct result (probability of at least two tails) is 0.8208.

What does the binomial distribution say about your problem?
 
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What is the definition of probability?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

What is an unfair coin?

An unfair coin is a coin that does not have an equal chance of landing on either heads or tails. This can be due to imbalances in weight or shape of the coin, or external factors such as the way it is flipped or the surface it lands on.

How can you calculate the probability of an unfair coin flip?

The probability of an unfair coin flip can be calculated by dividing the number of desired outcomes (such as heads) by the total number of possible outcomes. For example, if a coin is known to land on heads 60% of the time, the probability of getting heads on a single flip would be 0.6.

Can you have a probability greater than 1 or less than 0?

No, probabilities must always be between 0 and 1. A probability greater than 1 would indicate that an event is certain to occur, while a probability less than 0 would be impossible.

How do you account for bias in an unfair coin?

Bias in an unfair coin can be accounted for by adjusting the probability calculation. For example, if a coin is known to land on heads 60% of the time, but is biased towards heads and actually lands on heads 70% of the time, the probability of getting heads on a single flip would be adjusted to 0.7.

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