Conditional Probability

Conditional Probability

What is conditional probability?

• Conditional probability refers to the probability of an event occurring given that another event has already happened.
• An example is drawing names from a hat without replacement:
  • The probability of drawing a specific name first is 1/10.
  • The probability of drawing a specific name second is 1/9, unless it was already drawn first, in which case it is 0.
• Conditional probabilities change based on prior events.
• These probabilities can be illustrated using Venn diagrams, tree diagrams, or two-way tables.
• However, diagrams are not always necessary and can sometimes be bypassed for simpler questions.
• Conditional probability questions often include phrases like "given that".
  • Example: Finding the probability of rain today given that it rained yesterday.
• The "given that" context can affect the probability.
• Conditional probabilities are sometimes denoted using the notation 𝑃(𝐴∣𝐵)P(A∣B), read as "the probability of A given B".
  • Example: P(passes∣no revision) indicates the probability of passing given no revision, which differs from P(passes∣lots of revision).

Combined Conditional Probabilities

What are combined conditional probabilities?

• Trickier probability questions often involve both conditional and combined probabilities.
• An example is drawing multiple counters of different colors from a bag without replacement.
• These are known as 'without replacement' questions.
• Multiple draws require combined probabilities to determine outcomes, such as drawing two counters of the same color.
• The outcome of each draw affects the probabilities of subsequent draws, necessitating the use of conditional probability to find the answers.

How do I answer combined conditional probability questions?

• Combine ideas of combined and conditional probability appropriately.
• Use the AND (×) and OR (+) rules from combined probability.
• Remember, events must be independent to use AND, and mutually exclusive to use OR.
• Probabilities for subsequent events change based on conditional probability.
• Example: Drawing counters from a bag without replacement changes probabilities for each draw.
• Consider a bag with 7 green and 3 purple counters.
• Draw one counter, record its color, then draw a second without replacing the first.
• Probability both counters are green:
  • Probability the first counter is green: 7/10.
  • If the first is green, probability the second is green: 6/9.
• This demonstrates the conditional probability concept.
• To find 𝑃(both green)P(both green), multiply the two probabilities:
  • 
• This demonstrates the combined probability concept.

What about when two things are happening at the same time?

• Sometimes a question may not explicitly state the sequence of events.
• For example, it might simply say that two counters are drawn from a bag, without mentioning the order.
• You can assume that events happen one after the other for simplicity.
• This assumption does not change the mathematics of the question but makes it easier to solve.
• For example, drawing two counters from a bag can be treated as drawing one counter and then a second without replacement.
• This allows you to break down the problem into finding the probability of the 'first counter' and the 'second counter'.
• Drawing two counters simultaneously is mathematically equivalent to drawing them sequentially without replacement.

Are there any useful shortcuts for combined conditional probability questions?

• Consider a bag with 7 green counters and 3 purple counters. If 2 counters are drawn without replacement, we can determine the probability of the two counters being of different colors using the AND/OR rule. 
  • This involves considering '[1st green AND 2nd purple] OR [1st purple AND 2nd green]' to calculate the desired probability.
• Considering a bag with 7 green counters and 3 purple counters.
• Drawing 2 counters without replacement.
• Utilizing the AND/OR rule - '[1st green AND 2nd purple] OR [1st purple AND 2nd green]' - for calculating the probability of two counters being of different colors.

Understanding Probabilities: A Simplified Explanation

• Consider a bag with 7 green and 3 purple counters, and 2 counters are drawn without replacement.
• To find the probability of drawing counters of different colors, use the AND/OR rule:
  • 
  • 
• Notice both AND probabilities are equal (21/90).
• This is because the numerators and denominators are the same but swapped.
• Another way:
  • There are two ways for different colors (green then purple, or purple then green).
  • Each way has the same probability
  • Therefore, the probability is
• This method is useful for more complex scenarios.
• Example: Probability of drawing 2 green and 2 purple counters from the same bag when drawing 4 counters without replacement:
  • Using the AND/OR method requires considering multiple sequences, like OR ,etc.
  • There are six such sequences to consider.
• Simplified method:
  • There are 6 ways for '2 green and 2 purple' to occur (GGPP, GPGP, GPPG, PPGG, PGPG, PGGP).
  • Each way has the same probability:
  • Therefore, the probability is