Apprendre Conditional Probability

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Conditional probability measures the likelihood of an event occurring, given that another event has already happened. This is a crucial concept in probability theory because many real-world situations involve events that are not independent. For example, if you draw two cards from a deck without replacement, the probability that the second card is an ace changes depending on whether the first card was an ace.

The formula for conditional probability is:

P(A|B) = \frac{P(A \cap B)}{P(B)}

where:

$P(A|B)$ is the probability of event $A$ occurring given that event $B$ has occurred;
$P(A ∩ B)$ is the probability that both events A and B occur;
$P(B)$ is the probability that event B occurs.

Example 1: drawing cards

Suppose you have a standard 52-card deck. What is the probability that the second card drawn is an ace, given that the first card drawn was an ace (and cards are not replaced)?

Let $A$ be the event "second card is an ace," and $B$ be the event "first card is an ace."

There are 4 aces in the deck;
If the first card is an ace, there are now 3 aces left and 51 cards remaining.

So, the conditional probability is:

P(\text{second is ace} | \text{first is ace}) = \frac{3}{51}

Example 2: rolling dice

Suppose you roll two six-sided dice. What is the probability that the sum is 9, given that at least one die shows a 4?

Let $A$ be "sum is 9";
Let $B$ be "at least one die is a 4."

First, count the total outcomes where at least one die is a 4:

$(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)$ ;
$(1,4), (2,4), (3,4), (5,4), (6,4)$ .

That is 11 outcomes (note $(4,4)$ is only counted once).

Now, how many of these have a sum of 9?

$(4,5)$ and $(5,4)$ .

So, the conditional probability is:

P(\text{sum is 9} | \text{at least one die is 4}) = \frac{2}{11}


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# Calculate conditional probability in Python

# Example: Probability that the second card is an ace, given the first card is an ace

# Total number of aces in a deck
total_aces = 4
# Total number of cards
total_cards = 52

# After drawing one ace, remaining aces and cards
remaining_aces = total_aces - 1
remaining_cards = total_cards - 1

# Conditional probability: second card is ace given first is ace
conditional_prob = remaining_aces / remaining_cards

print(f"Probability that the second card is an ace given the first is an ace: {conditional_prob:.4f}")

# Example: Probability that the sum is 9 given at least one die is a 4

# List all (die1, die2) pairs where at least one die is a 4
outcomes = []
for die1 in range(1, 7):
    for die2 in range(1, 7):
        if die1 == 4 or die2 == 4:
            outcomes.append((die1, die2))

# Count outcomes where sum is 9
favorable = [pair for pair in outcomes if sum(pair) == 9]

conditional_prob_dice = len(favorable) / len(outcomes)
print(f"Probability that the sum is 9 given at least one die is a 4: {conditional_prob_dice:.4f}")

A box contains 3 red balls and 2 blue balls. You draw two balls one after another without replacement. What is the correct expression for the probability that the second ball is blue, given that the first ball was red?

Sélectionnez la réponse correcte

P(\text{second is blue} | \text{first is red}) = P(\text{second is blue}) * P(\text{first is red})

P(\text{second is blue} | \text{first is red}) = \frac{P(\text{second is blue and first is red})}{P(\text{first is red})}

P(\text{second is blue} | \text{first is red}) = P(\text{second is blue and first is red}) * P(\text{first is red})

P(\text{second is blue} | \text{first is red}) = \frac{P(\text{first is red})}{P(\text{second is blue})}

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Section 1. Chapitre 5

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Section 1. Chapitre 5