L05 - Two-Step Experiments

Two-Step Experiments

Information

• A two-step experiment can be visualised using a tree diagram. You can find an example on OneNote.
• In a tree diagram, it starts from “Start” and each step is a column of outcomes.
• The sample space is composed of the combined probabilities of each result.
  • For example, in an experiment that flips two coins, $P(H,H)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}$

With Replacement

• Two-step experiments can have outcomes that can be replaced.
• This means that an outcome can be repeated.
• The probability of the outcomes will also stay the same.

KICK (With Replacement)

The letters are chosen from the word KICK.

• (A tree diagram can be found on OneNote.)
• In a scenario where the experiment is with replacement, the probabilities will not change.
• The sample space will be {KK, KI, KC, IK, II, IC, CK, CI, CC}.
  • The probability of K is $\frac{2}{4}$.
  • The probability of I is $\frac{1}{4}$.
  • The probability of C is $\frac{1}{4}$.
• Here’s a few probability questions based on this experiment.
  1. $P(\boxed{K\; then\; C})=\frac{2}{4}\times \frac{1}{4}=\frac{1}{8}$
  2. $P(\boxed{a\; K\; and\; a\; C})=P(K,C)+P(C,K)$
     • $\frac{2}{4}\times \frac{1}{4}+\frac{1}{4}\times \frac{2}{4}=\frac{1}{4}$
  3. $P(\boxed{K,\; K\; |\; at\; least\; one\; K\; (given)})=\frac{n(K,K)}{n(\boxed{at\; least\; one\; K})}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$

Without Replacement

• Two step experiments can have outcomes that cannot be replaced.
• This means that an outcome cannot be repeated.
• The probability of outcomes will change depending on the remaining outcomes.

KICK (Without Replacement)

The letters are chosen from the word KICK.

• (A tree diagram can be found on OneNote.)
• In a scenario where the experiment is with replacement, the probabilities will not change.
• The sample space will be {KK, KI, KC, IK, II, IC, CK, CI, CC}.
  • In the first step, the outcomes of K, I, C are the same as the example With Replacement.
  • In the second step, the outcomes of K, I, and C are reduced.
    • If the outcome of the first step was K …
      • The probability of K is now $\frac{1}{3}$.
      • The probability of I is now $\frac{1}{3}$.
      • The probability of C is now $\frac{1}{3}$.
    • If the outcome of the first step was I …
      • The probability of K is now $\frac{2}{3}$.
      • The probability of C is now $\frac{1}{3}$.
    • If the outcome of the first step was C …
      • The probability of K is now $\frac{2}{3}$.
      • The probability of I is now $\frac{1}{3}$.
  • To find the probability of the results, you must combine the outcomes of the two steps. Refer to the initial information.
• Here’s a few probability questions based on this experiment.
  1. $P(\boxed{K\; then\; C})=\frac{1}{6}$
  2. $P(\boxed{a\; K\; and\; a\; C})=\frac{1}{6}+\frac{1}{6}=\frac{1}{3}$
  3. $P(\boxed{KK\; |\; at\; least\; one\; K})=\frac{P(K,K)}{P(\boxed{at\; least\; one\; K})}=\frac{\frac{1}{6}}{\frac{5}{6}}=\frac{1}{5}$
     • To get $\frac{5}{6}$, you combine the outcomes of KK, KI, KC, IK, and CK.