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 3 times and lost 7 times.

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

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 358 times and lost 642 times.

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