Download IB Math SL 23CD notes

23CD
C: Expectation
D: The Binomial Distribution
23C: EXPECTATION
• If there are n trials of an experiment, and an event has probability p of
occurring in each of the trials, then the number of times we expect the event
to occur is np.
• The expected outcome for the random variable X is the mean result 𝜇.
• In general, the expectation of the random variable X is
TWO EXAMPLES:
23C: FAIR GAMES
• In gambling, we say the expected gain of the player from each game is the
expected return or payout from the game, less the amount it cost them to
play.
• The game will be fair if the expected gain is zero.
23C: ANOTHER EXAMPLE
23D: THE BINOMIAL DISTRIBUTION
• Special type of discrete random variable which is applied to sampling with
replacement.
• The probability distribution that is associated with this variable is binomial
probability distribution.
23D: BINOMIAL EXPERIMENTS
23D: EXAMPLE
• Suppose a spinner has three blue edges and one white edge.
• The chance of finishing on blue is ¾ and on white is ¼.
• If we call a blue result a “success” and white a “failure”, then we have a
binomial experiment. WOW!
• Consider twirling the spinner n = 3 times. This is exciting.
• Find P(X = 0), P(X = 1), P(X = 2), and P(X = 3).
23D: EXAMPLE
• Suppose a spinner has three blue edges and one white edge.
• Consider twirling the spinner n = 3 times. This is still exciting.
• Find P(X = 0), P(X = 1), P(X = 2), and P(X = 3).
DID YOU NOTICE?
MORE FUN EXAMPLES FROM 23D
23D: THE BINOMIAL PROBABILITY DISTRIBUTION FUNCTION
EXAMPLE WITH CALCULATOR
EXAMPLE WITH CALCULATOR CONTINUED
23D: THE MEAN AND STANDARD DEVIATION OF A
BINOMIAL DISTRIBUTION
23D: THE MEAN AND STANDARD DEVIATION OF A
BINOMIAL DISTRIBUTION EXAMPLE