- Marginal_probability_Conditional_probability_Joint_probability_Independent_trial_Law_of_the_total_probability ================================================================================ Marginal probability: probability value which is assinged into one event without conditions ================================================================================ 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 ================================================================================ 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 $$$P[A\cap B] = P[B] \times P[A]$$$ ================================================================================ 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, $$$(A\cap B_1) \cap (A\cap B_2) \cup \cdots \cup (A\cap B_N)$$$ ================================================================================ 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. ================================================================================