- Marginal_probability_Conditional_probability_Joint_probability_Independent_trial_Law_of_the_total_probability
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Marginal probability: probability value which is assinged into one event without conditions
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Conditional probability
- Suppose there are 2 events A and B
- When probability value of event B occuring is known,
you can calculate probability of event A occuring in the situation of B occurred.
- $$$P[A|B]$$$
Under constraint of $$$P[B]>0$$$,
$$$P[A|B]=\dfrac{P[A\cup B]}{P[B]}$$$
- It's like a sample space becomes B
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Joint probability
- You can get following equation from conditional probability.
$$$P[B]P[A|B]=P[A\cap B]$$$ or $$$P[A]P[B|A]=P[A\cap B]$$$
- Joint probability is the probability of both A and B occuring
- Independent trial: trial where each trial is independent to others like rolling a dice twice.
If A and B are independent events, $$$P[A|B]=P[A]$$$
- Therefore, you can write $$$P[A\cap B] = P[B] \times P[A]$$$
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Law of the total probability
- Suppose sample space is composed of $$$B_1, B_2, \cdots, B_N$$$
- Suppose all Bs are not overlapped.
- Suppose event A is located like above.
Since A is entirely involved in S, A can be written as $$$a=A\cap S$$$
- You already supposed S is composed of $$$B_1 \cup B_2 \cup \cdots \cup B_N$$$
Therefore, you can write,
$$$A \cap (B_1 \cup B_2 \cup \cdots \cup B_N)$$$
- And you can write this,
$$$(A\cap B_1) \cap (A\cap B_2) \cup \cdots \cup (A\cap B_N)$$$
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You can write above concept in terms of probability.
- $$$P[A]=P[A\cap B_1]+P[A\cap B_2]+\cdots+P[A\cap B_N]$$$
- You can use $$$P[A\cap B]=P[A|B]P[B]$$$ on above notation.
$$$P[A]=P[A\cap B_1]=P[A|B_1]P[B_1] + \cdots + P[A|B_N]P[B_N]$$$
- You can call above notation as the total probability of event A.
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