# S1 Probability Coin Toss

• AntSC
In summary, the probability of getting two heads and one tail in any order is:3\left ( H \right )^{2}\left ( T \right )
AntSC
Having trouble with certain binomial and geometric distribution questions, which is indicating that my understanding isn't completely there yet. Any help would be greatly appreciated.

1. Homework Statement

A bag contains two biased coins: coin A shows Heads with a probability of 0.6, and coin B shows Heads with a probability 0.25. A coin is chosen at random from the bag and tossed three times.
Find the probability that the three tosses of the coin show two Heads and one Tail in any order.

## The Attempt at a Solution

Probabilities:
$H_{A}=0.6$ and $T_{A}=0.4$
$H_{B}=0.25$ and $T_{B}=0.75$

Possibilities for 2 heads and one tail in any order:
$$3\left ( H \right )^{2}\left ( T \right )$$

Is this correct so far?
My question is how to incorporate the probability of picking coin A or coin B into the problem?

What is the probability you picked coin A?

A half

So if you picked a coin at random and tossed just once, what is the probability the result is a head?

Ah i see it now.
$$P=\frac{1}{2}3\left ( H_{A} \right )^{2}\left ( T_{A} \right )+\frac{1}{2}3\left ( H_{B} \right )^{2}\left ( T_{B} \right )$$
Is this right?

You can check it with a probability tree if you are unsure.

Sure. I want to start to dispense with the need for visual aids and make sure i can construct the problem without.
Especially when dealing with a larger set of choices, like 52 cards. A tree then won't be so helpful.
Thanks for the dialogue. I think i needed to get it out there to help work it through.
You might see a few more questions from me in future :)

AntSC said:
Ah i see it now.
$$P=\frac{1}{2}3\left ( H_{A} \right )^{2}\left ( T_{A} \right )+\frac{1}{2}3\left ( H_{B} \right )^{2}\left ( T_{B} \right )$$
Is this right?

QUOTE="AntSC, post: 5021860, member: 450435"]Ah i see it now.
$$P=\frac{1}{2}3\left ( H_{A} \right )^{2}\left ( T_{A} \right )+\frac{1}{2}3\left ( H_{B} \right )^{2}\left ( T_{B} \right )$$
Is this right?[/QUOTE]

If ##E## is the event "2H, 1T (any order)", does your formula satisfy the basic relationship
$$P(E) = P(E|A) P(A) + P(E|B) P(B) ?$$
If it does, it is OK.

BTW: you might compare this with the scenario where you replace the coin after each toss and then ask about ##E##.

## 1. What is the probability of getting heads or tails on a coin toss?

The probability of getting either heads or tails on a coin toss is 50%, assuming the coin is fair and has equal chances of landing on either side.

## 2. Can the outcome of a coin toss be predicted?

No, the outcome of a coin toss is random and cannot be predicted with certainty. The probability of getting heads or tails is always 50% regardless of past outcomes.

## 3. How does the sample size affect the probability of getting heads or tails?

The sample size does not affect the probability of getting heads or tails on a coin toss. Each toss is an independent event and the probability remains at 50% regardless of the number of tosses.

## 4. What is the difference between theoretical probability and experimental probability?

Theoretical probability is the expected probability of an event based on mathematical calculations, while experimental probability is the probability based on actual experiments or observations. In the case of a coin toss, theoretical probability would be 50%, while experimental probability could vary depending on the number of tosses.

## 5. How can probability be useful in everyday life?

Probability can be used to make informed decisions based on the likelihood of certain events occurring. For example, businesses may use probability to determine the success of a new product, while individuals may use it to make financial decisions or assess risks.

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