Question

Soil alkalization can severely affect crop growth. One study decided to investigate the affects of sodium carbonate (an alkaline substance) on a specific variety of grass. Three biological replicates of the grass were created and each replicate was treated with a different amount of the sodium carbonate. To measure the effect of each treatment, proteins were measured in each of the three replicates. Suppose that 92 proteins were present in Replicate I, 103 proteins were present in Replicate II, and 97 proteins were present in Replicate III. Of those proteins, 36 were present in both Replicate I and II, 41 were present in Replicate I and III, 39 were present in Replicate II and III, and 24 were present in all three replicates.

a. How many different proteins were present in at least one of the three replicates?

b. A protein is selected at random. What is the probability that it is present in exactly two of the replicates?

Answer 1

• ANSWER::

  (a)

  Let R1, R2 and R3 denote the event that protein is present in Replicate I , II and III respectively.

  Number of proteins present in at least one of the replicates is

  n(R1 $\cup$ R2 $\cup$ R3) = n(R1) + n(R2) + n(R3) - n(R1 $\cap$ R2) - n(R1 $\cap$ R3) - n(R2 $\cap$ R3) +  n(R1 $\cap$ R2 $\cap$ R3)

  = 92 + 103 + 97 - 36 - 41 - 39 + 24

  = 200

  (b)
  Number of proteins present in exactly two replicates

  = n(R1 $\cap$ R2) + n(R1 $\cap$ R3) + n(R2 $\cap$ R3) - 2 * n(R1 $\cap$ R2 $\cap$ R3)

  = 36 + 41 + 39 - 2 *  24

  = 68

  Probability that a random protein is present in exactly two replicates = Number of proteins present in exactly two replicates / Total number of proteins

  = 68 / 200

  = 0.34

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