Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
7. Goat
8. Car
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 333 times and lost 667 times.

1. Goat
2. Goat
3. Car
4. Car
5. Goat
6. Car
7. Goat
8. Car
9. Car
10. Goat
11. Car
12. Car
13. Car
14. Car
15. Car
16. Car
17. Goat
18. Goat
19. Goat
20. Car
21. Goat
22. Goat
23. Goat
24. Car
25. Car
26. Goat
27. Car
28. Goat
29. Car
30. Car
31. Goat
32. Goat
33. Goat
34. Goat
35. Goat
36. Car
37. Goat
38. Car
39. Goat
40. Goat
41. Goat
42. Goat
43. Car
44. Goat
45. Goat
46. Goat
47. Car
48. Car
49. Goat
50. Goat
51. Goat
52. Goat
53. Car
54. Car
55. Goat
56. Goat
57. Car
58. Goat
59. Goat
60. Car
61. Goat
62. Goat
63. Car
64. Car
65. Goat
66. Goat
67. Goat
68. Goat
69. Goat
70. Goat
71. Goat
72. Goat
73. Goat
74. Goat
75. Goat
76. Goat
77. Goat
78. Goat
79. Car
80. Car
81. Goat
82. Car
83. Goat
84. Car
85. Goat
86. Goat
87. Goat
88. Goat
89. Car
90. Goat
91. Car
92. Car
93. Goat
94. Car
95. Car
96. Car
97. Goat
98. Goat
99. Goat
100. Car
101. Goat
102. Goat
103. Car
104. Goat
105. Car
106. Goat
107. Car
108. Goat
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Goat
115. Goat
116. Goat
117. Goat
118. Goat
119. Car
120. Car
121. Goat
122. Goat
123. Goat
124. Goat
125. Car
126. Goat
127. Car
128. Goat
129. Goat
130. Car
131. Car
132. Car
133. Car
134. Goat
135. Goat
136. Goat
137. Goat
138. Car
139. Goat
140. Goat
141. Goat
142. Car
143. Goat
144. Goat
145. Goat
146. Goat
147. Goat
148. Goat
149. Goat
150. Car
151. Car
152. Goat
153. Goat
154. Goat
155. Car
156. Goat
157. Car
158. Goat
159. Car
160. Goat
161. Goat
162. Goat
163. Car
164. Car
165. Goat
166. Car
167. Goat
168. Goat
169. Goat
170. Goat
171. Goat
172. Goat
173. Goat
174. Goat
175. Car
176. Car
177. Car
178. Goat
179. Goat
180. Goat
181. Car
182. Goat
183. Goat
184. Goat
185. Car
186. Car
187. Goat
188. Goat
189. Goat
190. Goat
191. Goat
192. Goat
193. Car
194. Goat
195. Goat
196. Goat
197. Goat
198. Car
199. Car
200. Goat
201. Goat
202. Goat
203. Goat
204. Goat
205. Goat
206. Car
207. Goat
208. Goat
209. Goat
210. Goat
211. Car
212. Car
213. Goat
214. Goat
215. Goat
216. Goat
217. Goat
218. Goat
219. Car
220. Goat
221. Goat
222. Car
223. Car
224. Goat
225. Goat
226. Goat
227. Goat
228. Car
229. Car
230. Goat
231. Goat
232. Goat
233. Car
234. Car
235. Car
236. Goat
237. Goat
238. Car
239. Goat
240. Goat
241. Goat
242. Car
243. Car
244. Goat
245. Car
246. Car
247. Car
248. Goat
249. Goat
250. Goat
251. Car
252. Car
253. Car
254. Car
255. Car
256. Goat
257. Car
258. Goat
259. Car
260. Goat
261. Goat
262. Goat
263. Goat
264. Car
265. Car
266. Goat
267. Goat
268. Goat
269. Goat
270. Car
271. Goat
272. Goat
273. Car
274. Goat
275. Goat
276. Goat
277. Car
278. Goat
279. Goat
280. Goat
281. Car
282. Goat
283. Goat
284. Goat
285. Car
286. Goat
287. Car
288. Car
289. Goat
290. Goat
291. Goat
292. Car
293. Goat
294. Goat
295. Goat
296. Goat
297. Goat
298. Goat
299. Goat
300. Car
301. Goat
302. Car
303. Goat
304. Car
305. Goat
306. Goat
307. Goat
308. Goat
309. Goat
310. Goat
311. Goat
312. Goat
313. Car
314. Car
315. Goat
316. Goat
317. Goat
318. Goat
319. Car
320. Goat
321. Goat
322. Car
323. Car
324. Goat
325. Car
326. Car
327. Goat
328. Goat
329. Goat
330. Car
331. Goat
332. Car
333. Goat
334. Goat
335. Car
336. Goat
337. Car
338. Goat
339. Car
340. Goat
341. Goat
342. Goat
343. Goat
344. Goat
345. Goat
346. Goat
347. Car
348. Goat
349. Goat
350. Goat
351. Goat
352. Goat
353. Car
354. Goat
355. Goat
356. Goat
357. Car
358. Goat
359. Goat
360. Goat
361. Goat
362. Car
363. Goat
364. Car
365. Car
366. Goat
367. Goat
368. Goat
369. Car
370. Goat
371. Goat
372. Goat
373. Goat
374. Goat
375. Car
376. Goat
377. Goat
378. Goat
379. Car
380. Car
381. Car
382. Car
383. Goat
384. Car
385. Goat
386. Car
387. Goat
388. Goat
389. Goat
390. Goat
391. Goat
392. Goat
393. Goat
394. Goat
395. Goat
396. Car
397. Goat
398. Goat
399. Car
400. Goat
401. Goat
402. Goat
403. Car
404. Goat
405. Goat
406. Goat
407. Goat
408. Car
409. Goat
410. Goat
411. Car
412. Goat
413. Goat
414. Car
415. Car
416. Goat
417. Goat
418. Goat
419. Goat
420. Goat
421. Goat
422. Goat
423. Goat
424. Car
425. Car
426. Goat
427. Car
428. Car
429. Goat
430. Goat
431. Goat
432. Goat
433. Car
434. Goat
435. Goat
436. Goat
437. Car
438. Goat
439. Goat
440. Goat
441. Goat
442. Car
443. Car
444. Goat
445. Goat
446. Goat
447. Car
448. Goat
449. Goat
450. Goat
451. Goat
452. Goat
453. Goat
454. Car
455. Car
456. Goat
457. Goat
458. Goat
459. Goat
460. Car
461. Goat
462. Goat
463. Goat
464. Goat
465. Car
466. Goat
467. Car
468. Car
469. Car
470. Car
471. Car
472. Car
473. Goat
474. Goat
475. Car
476. Goat
477. Goat
478. Goat
479. Car
480. Car
481. Car
482. Goat
483. Goat
484. Goat
485. Car
486. Goat
487. Goat
488. Goat
489. Goat
490. Car
491. Goat
492. Goat
493. Goat
494. Car
495. Goat
496. Goat
497. Car
498. Goat
499. Car
500. Goat
501. Car
502. Goat
503. Car
504. Car
505. Goat
506. Goat
507. Car
508. Car
509. Car
510. Car
511. Goat
512. Goat
513. Car
514. Goat
515. Goat
516. Car
517. Goat
518. Car
519. Goat
520. Goat
521. Goat
522. Car
523. Goat
524. Goat
525. Goat
526. Goat
527. Goat
528. Goat
529. Goat
530. Goat
531. Car
532. Goat
533. Goat
534. Goat
535. Goat
536. Car
537. Goat
538. Goat
539. Goat
540. Goat
541. Goat
542. Goat
543. Goat
544. Goat
545. Goat
546. Goat
547. Goat
548. Goat
549. Goat
550. Car
551. Car
552. Goat
553. Car
554. Goat
555. Goat
556. Car
557. Goat
558. Goat
559. Car
560. Goat
561. Car
562. Goat
563. Goat
564. Car
565. Car
566. Goat
567. Car
568. Goat
569. Goat
570. Car
571. Car
572. Car
573. Car
574. Goat
575. Car
576. Goat
577. Car
578. Car
579. Goat
580. Car
581. Car
582. Goat
583. Goat
584. Car
585. Goat
586. Goat
587. Goat
588. Goat
589. Goat
590. Car
591. Goat
592. Goat
593. Car
594. Goat
595. Car
596. Goat
597. Car
598. Goat
599. Car
600. Goat
601. Car
602. Goat
603. Goat
604. Goat
605. Goat
606. Car
607. Car
608. Goat
609. Goat
610. Goat
611. Goat
612. Car
613. Goat
614. Car
615. Goat
616. Goat
617. Goat
618. Goat
619. Goat
620. Goat
621. Car
622. Car
623. Car
624. Goat
625. Car
626. Car
627. Goat
628. Goat
629. Goat
630. Goat
631. Goat
632. Goat
633. Goat
634. Goat
635. Car
636. Goat
637. Car
638. Goat
639. Car
640. Goat
641. Goat
642. Goat
643. Goat
644. Car
645. Goat
646. Goat
647. Car
648. Car
649. Goat
650. Goat
651. Goat
652. Goat
653. Goat
654. Goat
655. Goat
656. Car
657. Goat
658. Car
659. Goat
660. Goat
661. Car
662. Car
663. Goat
664. Goat
665. Car
666. Goat
667. Goat
668. Car
669. Car
670. Goat
671. Goat
672. Car
673. Goat
674. Goat
675. Goat
676. Goat
677. Goat
678. Goat
679. Car
680. Goat
681. Goat
682. Goat
683. Goat
684. Goat
685. Goat
686. Goat
687. Goat
688. Goat
689. Car
690. Goat
691. Goat
692. Goat
693. Goat
694. Goat
695. Car
696. Goat
697. Car
698. Goat
699. Goat
700. Car
701. Car
702. Goat
703. Goat
704. Goat
705. Goat
706. Goat
707. Car
708. Goat
709. Goat
710. Car
711. Car
712. Goat
713. Car
714. Goat
715. Car
716. Goat
717. Goat
718. Car
719. Car
720. Goat
721. Goat
722. Car
723. Goat
724. Goat
725. Goat
726. Car
727. Goat
728. Car
729. Goat
730. Goat
731. Goat
732. Goat
733. Goat
734. Goat
735. Goat
736. Goat
737. Goat
738. Goat
739. Goat
740. Goat
741. Car
742. Goat
743. Car
744. Car
745. Car
746. Car
747. Car
748. Goat
749. Car
750. Goat
751. Goat
752. Car
753. Car
754. Car
755. Goat
756. Goat
757. Car
758. Goat
759. Goat
760. Car
761. Goat
762. Goat
763. Car
764. Goat
765. Goat
766. Goat
767. Car
768. Car
769. Car
770. Car
771. Goat
772. Goat
773. Goat
774. Goat
775. Car
776. Goat
777. Car
778. Car
779. Car
780. Car
781. Car
782. Goat
783. Goat
784. Goat
785. Goat
786. Goat
787. Goat
788. Goat
789. Goat
790. Goat
791. Car
792. Car
793. Car
794. Goat
795. Car
796. Car
797. Goat
798. Goat
799. Car
800. Goat
801. Car
802. Car
803. Goat
804. Goat
805. Car
806. Goat
807. Goat
808. Goat
809. Goat
810. Car
811. Goat
812. Car
813. Goat
814. Car
815. Goat
816. Goat
817. Car
818. Goat
819. Goat
820. Goat
821. Goat
822. Goat
823. Goat
824. Goat
825. Goat
826. Goat
827. Car
828. Goat
829. Car
830. Car
831. Goat
832. Goat
833. Car
834. Car
835. Goat
836. Goat
837. Goat
838. Goat
839. Goat
840. Goat
841. Car
842. Goat
843. Goat
844. Car
845. Goat
846. Goat
847. Goat
848. Car
849. Goat
850. Goat
851. Car
852. Car
853. Goat
854. Goat
855. Car
856. Goat
857. Goat
858. Car
859. Goat
860. Goat
861. Car
862. Car
863. Goat
864. Goat
865. Car
866. Goat
867. Goat
868. Goat
869. Goat
870. Goat
871. Goat
872. Goat
873. Goat
874. Car
875. Goat
876. Goat
877. Car
878. Goat
879. Goat
880. Car
881. Goat
882. Goat
883. Goat
884. Car
885. Car
886. Goat
887. Goat
888. Car
889. Goat
890. Car
891. Goat
892. Goat
893. Goat
894. Car
895. Car
896. Goat
897. Goat
898. Goat
899. Goat
900. Goat
901. Goat
902. Goat
903. Goat
904. Goat
905. Goat
906. Goat
907. Car
908. Goat
909. Goat
910. Car
911. Goat
912. Goat
913. Goat
914. Car
915. Goat
916. Goat
917. Car
918. Goat
919. Goat
920. Goat
921. Car
922. Goat
923. Goat
924. Car
925. Goat
926. Goat
927. Goat
928. Car
929. Goat
930. Goat
931. Car
932. Goat
933. Goat
934. Goat
935. Goat
936. Goat
937. Goat
938. Goat
939. Goat
940. Car
941. Car
942. Goat
943. Car
944. Goat
945. Goat
946. Goat
947. Goat
948. Goat
949. Car
950. Goat
951. Goat
952. Goat
953. Car
954. Goat
955. Goat
956. Goat
957. Goat
958. Car
959. Car
960. Goat
961. Goat
962. Car
963. Goat
964. Goat
965. Goat
966. Goat
967. Car
968. Goat
969. Goat
970. Goat
971. Goat
972. Car
973. Goat
974. Goat
975. Goat
976. Goat
977. Car
978. Goat
979. Goat
980. Car
981. Goat
982. Car
983. Goat
984. Car
985. Goat
986. Goat
987. Goat
988. Car
989. Goat
990. Goat
991. Goat
992. Car
993. Car
994. Car
995. Goat
996. Car
997. Car
998. Car
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 90 times and lost 910 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 34 times and lost 66 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}